IODS course project
Donggyu Kam
About the project
I am so excited to get an opportunity to be into R. I am expecting to learn how to hand data and visulaze the results of analysis through R. One of my colleagues had let me know about this course.
The URL for my git repository as follow:
https://github.com/Donggyu-Kam/IODS-project
Regression and Model validation (Excercise 2)
The data is the student’s attitude toward statistics.The data consists of information about students’ ages, students’ gender, three approaches, students’ attitude toward statistics and exam points. The oberservations whare the exam points are zeor were removed from data. And thus, the data has 166 observations and 7 variables. You can figure out three variables to affect the exam points through linear regression models and finally you will figure out the most significant variable to exampoints
#Before handling data, you should install two packages:“ggplot2” and“GGally”.
1.To read and explore data
- Read data and look into the structure and dimension of data
- df<-read.table(“http://s3.amazonaws.com/assets.datacamp.com/production/course_2218/datasets/learning2014.txt”, sep=“,”, header=TRUE )
- str(df)
- dim(df)
- Initialize plot with data and aesthetic mapping
- p1 <- ggplot(df, aes(x = attitude,col=gender, y = points))
- Define the visualization type (points)
- p2 <- p1 + geom_point()
- Draw the plot
- p2
- Add a regression line
- p3 <- p2 + geom_smooth(method = “lm”)
- Add a main title and draw the plot
- p4 <- p3+ggtitle(“<Student’s attitude versus exam points>”) &
- p4
2. To choose variables and create a regression model
By drawing plots illustrating the relationships between the variables and the normality of the erros distributions, it will find out what variables are closely related to exam points.
- Draw a scatter plot matrix of the variables in learning2014
- p <- ggpairs(df, mapping = aes(col=gender, alpha = 0.3), lower = list(combo = wrap(“facethist”, bins = 20)))
- Plotting
- p
According to the plot “p”, three variables “attitude”, “stra” and “surf” were chosen due to the strongest correlation value as explanatory variables.
- Fit a linear model to each variable
- qplot(attitude, points, data = df) + geom_smooth(method = “lm”)
- qplot(stra, points, data=df)+ geom_smooth(method =“lm”)
- qplot(surf, points, data=df)+ geom_smooth(method=“lm”)
After fitting each variable to a linear regression modle, “attitude” shows postive relationship but the others show week or negative relationships. So, the linear regression model is re-fitted with three variables together.
- Fit a linear modewl with three variables together
- mymodel<-lm(points~attitude+stra+surf, data=df)
- Take a look at the summary of the model
- summary(mymodel)
In this model, “Stra” and “Surf” are not not significantly related to the exam points. So, the model is re-fitted again witout the two variables
Fit the model again with the “attitude”
- mymodel2<- lm(points ~ attitude, data=df)
- summary(mymodel2)
- par(mfrow = c(2,2))
- plot(mymodel2, which=c(1,2,5)) - only take looks at the relation between residuals and fitted values, the normal QQplot, and the leverage of observations.
3. Result of regression model analysis
By analyzing the residuals and errors of the model, we can test the model validation. The variance of errors in residulas over fitted values looks constant without any patterns but there are 4 outliers indicating that the model only with “attitude” cannot explain the 4 ouliters of exam points. In the normality of errors in QQ plot, the errors are normaly distributed. In leverage of observations, standardized residuals are scattered around 0 and there seems no impact of outliers
Losistic Regression (Excercise 3)
The data is about student acheivement in secondary education of two Portugues schools. Two datasets are available for the performance in two subjects: Math and Portuguese language.
P. Cortez and A. Silva. Using Data Mining to Predict Secondary School Student Performance. In A. Brito and J. Teixeira Eds., Proceedings of 5th FUture BUsiness TEChnology Conference (FUBUTEC 2008) pp. 5-12, Porto, Portugal, April, 2008, EUROSIS, ISBN 978-9077381-39-7.[link]http://www3.dsi.uminho.pt/pcortez/student.pdf
1. Retreiving data
the data is about student acheivement in secondary education of two Portugues schools according to the student grades, demographic and school-related features. Two datasets are available for the performance in two subjects: Math and Portuguese language. the datasets consists of 35 variables and 382 observations.I firstly beging with installing some packages helps analysis. And thus, we can take a glimpse at the datasets and what variables are available through column names on the datasets.
library(dplyr);library(ggplot2);library(tidyr);library(boot)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionalc<-read.csv("Z:\\IODS-project\\data\\alc.csv")
glimpse(alc)## Observations: 382
## Variables: 36
## $ X <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...
## $ school <fct> GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP,...
## $ sex <fct> F, F, F, F, F, M, M, F, M, M, F, F, M, M, M, F, F, ...
## $ age <int> 18, 17, 15, 15, 16, 16, 16, 17, 15, 15, 15, 15, 15,...
## $ address <fct> U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, ...
## $ famsize <fct> GT3, GT3, LE3, GT3, GT3, LE3, LE3, GT3, LE3, GT3, G...
## $ Pstatus <fct> A, T, T, T, T, T, T, A, A, T, T, T, T, T, A, T, T, ...
## $ Medu <int> 4, 1, 1, 4, 3, 4, 2, 4, 3, 3, 4, 2, 4, 4, 2, 4, 4, ...
## $ Fedu <int> 4, 1, 1, 2, 3, 3, 2, 4, 2, 4, 4, 1, 4, 3, 2, 4, 4, ...
## $ Mjob <fct> at_home, at_home, at_home, health, other, services,...
## $ Fjob <fct> teacher, other, other, services, other, other, othe...
## $ reason <fct> course, course, other, home, home, reputation, home...
## $ nursery <fct> yes, no, yes, yes, yes, yes, yes, yes, yes, yes, ye...
## $ internet <fct> no, yes, yes, yes, no, yes, yes, no, yes, yes, yes,...
## $ guardian <fct> mother, father, mother, mother, father, mother, mot...
## $ traveltime <int> 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, ...
## $ studytime <int> 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 3, 1, 3, ...
## $ failures <int> 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
## $ schoolsup <fct> yes, no, yes, no, no, no, no, yes, no, no, no, no, ...
## $ famsup <fct> no, yes, no, yes, yes, yes, no, yes, yes, yes, yes,...
## $ paid <fct> no, no, yes, yes, yes, yes, no, no, yes, yes, yes, ...
## $ activities <fct> no, no, no, yes, no, yes, no, no, no, yes, no, yes,...
## $ higher <fct> yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, y...
## $ romantic <fct> no, no, no, yes, no, no, no, no, no, no, no, no, no...
## $ famrel <int> 4, 5, 4, 3, 4, 5, 4, 4, 4, 5, 3, 5, 4, 5, 4, 4, 3, ...
## $ freetime <int> 3, 3, 3, 2, 3, 4, 4, 1, 2, 5, 3, 2, 3, 4, 5, 4, 2, ...
## $ goout <int> 4, 3, 2, 2, 2, 2, 4, 4, 2, 1, 3, 2, 3, 3, 2, 4, 3, ...
## $ Dalc <int> 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
## $ Walc <int> 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1, 3, 2, 1, 2, 2, ...
## $ health <int> 3, 3, 3, 5, 5, 5, 3, 1, 1, 5, 2, 4, 5, 3, 3, 2, 2, ...
## $ absences <int> 6, 4, 10, 2, 4, 10, 0, 6, 0, 0, 0, 4, 2, 2, 0, 4, 6...
## $ G1 <int> 5, 5, 7, 15, 6, 15, 12, 6, 16, 14, 10, 10, 14, 10, ...
## $ G2 <int> 6, 5, 8, 14, 10, 15, 12, 5, 18, 15, 8, 12, 14, 10, ...
## $ G3 <int> 6, 6, 10, 15, 10, 15, 11, 6, 19, 15, 9, 12, 14, 11,...
## $ alc_use <dbl> 1.0, 1.0, 2.5, 1.0, 1.5, 1.5, 1.0, 1.0, 1.0, 1.0, 1...
## $ high_use <lgl> TRUE, TRUE, FALSE, TRUE, TRUE, TRUE, TRUE, TRUE, TR...gather(alc)%>% glimpse## Warning: attributes are not identical across measure variables;
## they will be dropped## Observations: 13,752
## Variables: 2
## $ key <chr> "X", "X", "X", "X", "X", "X", "X", "X", "X", "X", "X", "...
## $ value <chr> "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11",...gather(alc)%>%ggplot(aes(value))+facet_wrap("key", scales="free")+geom_bar()## Warning: attributes are not identical across measure variables;
## they will be dropped
2. Hypothesis
In order to study the relationship between high and low alcohol consumptions and variables, i am going to work folloing to hypothesis;
- Gender - Males consume more alcohol than females.
- G1 - First period grade decreases with alcohol consumption.
- G2 - Second period grade decreases with alcohol consumption.
- G3 - Third period grade decreases with alcohol consumption.
I will take a look at each relationship followring to the hypothesis one by one p<-ggpairs(alc, mapping = aes(col=sex, alpha = 0.3), lower = list(combo = wrap(“facethist”, bins = 20))) p
Hypothesis 1 - Gender
g1 <- ggplot(data = alc, aes(x = alc_use, fill = sex))
g1 + geom_bar()+facet_wrap("sex") + ggtitle("alohol consumption by sex")
g11<-ggplot(data=alc, aes(x=high_use, fill=sex))
g11+geom_bar()+facet_wrap("sex") + ggtitle("high and low alcohol consumption by sex")
Hypothesis 2 - G1
g2 <- ggplot(data=alc, aes(x=high_use, y= G1, col=sex))
g2+geom_boxplot()+ggtitle("G1 by alcohol consumption and sex")
Hypothesis 3 - G2
g3<-ggplot(alc, aes(x=high_use, y=G2, col=sex))
g3+geom_boxplot()+ggtitle("G2 by acohol consumption and sex")
Hypothesis 4 - Absences
g4<-ggplot(alc, aes(x=high_use, y=G3, col=sex))
g4+geom_boxplot()+ggtitle("G3 by alcohol consumption and sex")
Interim result
As a result of exploration of relationship with 4 variable to the high and low alcohol relationship, it seems that femailes consume more alcohol than males do. However, it is presumed that the more males consum alcohol, the less grades males achieve.
3.Lostistic regression
From now, in order to interpret relationship between variables presumably relating to alcohol consumptions and alcohol consumption, we will deal with logistaic regression.
m<-glm(high_use~sex+G1+G2+G3, data=alc, family="binomial")
summary(m)##
## Call:
## glm(formula = high_use ~ sex + G1 + G2 + G3, family = "binomial",
## data = alc)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.7050 -1.1393 0.8236 1.0275 1.4369
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.124551 0.377648 -0.330 0.742
## sexM -0.929752 0.215442 -4.316 1.59e-05 ***
## G1 0.100421 0.063556 1.580 0.114
## G2 0.005342 0.076244 0.070 0.944
## G3 -0.032884 0.053600 -0.614 0.540
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 524.94 on 381 degrees of freedom
## Residual deviance: 502.43 on 377 degrees of freedom
## AIC: 512.43
##
## Number of Fisher Scoring iterations: 4coef(m)## (Intercept) sexM G1 G2 G3
## -0.124550963 -0.929752246 0.100420779 0.005341903 -0.032884237OR<-coef(m)%>%exp
OR## (Intercept) sexM G1 G2 G3
## 0.8828933 0.3946515 1.1056360 1.0053562 0.9676506CI<-exp(confint(m))## Waiting for profiling to be done...CI## 2.5 % 97.5 %
## (Intercept) 0.4202206 1.8520181
## sexM 0.2575779 0.5999088
## G1 0.9774251 1.2553494
## G2 0.8649561 1.1681070
## G3 0.8686826 1.0737634cbind(OR, CI)## OR 2.5 % 97.5 %
## (Intercept) 0.8828933 0.4202206 1.8520181
## sexM 0.3946515 0.2575779 0.5999088
## G1 1.1056360 0.9774251 1.2553494
## G2 1.0053562 0.8649561 1.1681070
## G3 0.9676506 0.8686826 1.0737634The model indicates taht males are more significantly related to high alcohol consumptions.However, the other 3 variables are not significantly related to high alcohol consumptions. In odds rates, the odds rates for G2 and G3 are more than 1.
Prediction power
probabilities<-predict(m, type="response")
alc<-mutate(alc,probability=probabilities)
alc<-mutate(alc,prediction=probability>0.5)
select(alc,G2,G3, high_use, probability, prediction)%>% tail(10)## G2 G3 high_use probability prediction
## 373 5 0 FALSE 0.3953044 FALSE
## 374 5 5 FALSE 0.3567518 FALSE
## 375 9 10 FALSE 0.6454027 TRUE
## 376 5 6 FALSE 0.5762451 TRUE
## 377 5 0 TRUE 0.6468232 TRUE
## 378 9 8 TRUE 0.5898627 TRUE
## 379 5 0 TRUE 0.6235593 TRUE
## 380 5 0 TRUE 0.6235593 TRUE
## 381 16 16 FALSE 0.4777423 FALSE
## 382 12 10 FALSE 0.4466009 FALSE2x2 cross-tabulation and graphic representation
g<-ggplot(alc,(aes(x=probability, y=high_use, col=prediction)))
g+geom_point()
table(high_use=alc$high_use, prediction=alc$prediction)%>%prop.table()%>%addmargins()## prediction
## high_use FALSE TRUE Sum
## FALSE 0.2225131 0.2225131 0.4450262
## TRUE 0.1492147 0.4057592 0.5549738
## Sum 0.3717277 0.6282723 1.0000000Exploring the training error
loss_func<-function(class, prob){
n_wrong <-abs(class - prob) > 0.5
mean(n_wrong)
}
loss_func(class = alc$high_use, prob = alc$probability)## [1] 0.3717277Through exploring the training error, the incorrection of prediction is almost 50%.
10-fold cross-validation (Bonus task)
cv<-cv.glm(data=alc, cost=loss_func, glmfit = m, K=10)
cv$delta[1]## [1] 0.3795812Eventually, the prediction error with the 4 variables is 40%.
Clustering and classification (Exercise 4)
library(MASS);library(tidyverse);library(corrplot)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## select## -- Attaching packages -------------------------------------------------------------------------------------- tidyverse 1.2.1 --## v tibble 2.1.3 v purrr 0.3.3
## v readr 1.3.1 v stringr 1.4.0
## v tibble 2.1.3 v forcats 0.4.0## -- Conflicts ----------------------------------------------------------------------------------------- tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag() masks stats::lag()
## x MASS::select() masks dplyr::select()## corrplot 0.84 loadeddata("Boston")The data has 506 observations and 14 variables. Data is about housing values in suburbs of Boston.
Explort dataset
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00pairs(Boston)
Correlate all variables
cor_matrix<-cor(Boston)
cor_matrix%>%round(2)## crim zn indus chas nox rm age dis rad tax
## crim 1.00 -0.20 0.41 -0.06 0.42 -0.22 0.35 -0.38 0.63 0.58
## zn -0.20 1.00 -0.53 -0.04 -0.52 0.31 -0.57 0.66 -0.31 -0.31
## indus 0.41 -0.53 1.00 0.06 0.76 -0.39 0.64 -0.71 0.60 0.72
## chas -0.06 -0.04 0.06 1.00 0.09 0.09 0.09 -0.10 -0.01 -0.04
## nox 0.42 -0.52 0.76 0.09 1.00 -0.30 0.73 -0.77 0.61 0.67
## rm -0.22 0.31 -0.39 0.09 -0.30 1.00 -0.24 0.21 -0.21 -0.29
## age 0.35 -0.57 0.64 0.09 0.73 -0.24 1.00 -0.75 0.46 0.51
## dis -0.38 0.66 -0.71 -0.10 -0.77 0.21 -0.75 1.00 -0.49 -0.53
## rad 0.63 -0.31 0.60 -0.01 0.61 -0.21 0.46 -0.49 1.00 0.91
## tax 0.58 -0.31 0.72 -0.04 0.67 -0.29 0.51 -0.53 0.91 1.00
## ptratio 0.29 -0.39 0.38 -0.12 0.19 -0.36 0.26 -0.23 0.46 0.46
## black -0.39 0.18 -0.36 0.05 -0.38 0.13 -0.27 0.29 -0.44 -0.44
## lstat 0.46 -0.41 0.60 -0.05 0.59 -0.61 0.60 -0.50 0.49 0.54
## medv -0.39 0.36 -0.48 0.18 -0.43 0.70 -0.38 0.25 -0.38 -0.47
## ptratio black lstat medv
## crim 0.29 -0.39 0.46 -0.39
## zn -0.39 0.18 -0.41 0.36
## indus 0.38 -0.36 0.60 -0.48
## chas -0.12 0.05 -0.05 0.18
## nox 0.19 -0.38 0.59 -0.43
## rm -0.36 0.13 -0.61 0.70
## age 0.26 -0.27 0.60 -0.38
## dis -0.23 0.29 -0.50 0.25
## rad 0.46 -0.44 0.49 -0.38
## tax 0.46 -0.44 0.54 -0.47
## ptratio 1.00 -0.18 0.37 -0.51
## black -0.18 1.00 -0.37 0.33
## lstat 0.37 -0.37 1.00 -0.74
## medv -0.51 0.33 -0.74 1.00corrplot(cor_matrix, method="circle", type="upper", cl.pos="b", tl.pos="d", tl.cex=0.6)
There is strongly positive correlation bewtween “rad” and “tax” and stongly negative correlation between “age” and “dis” and between “lstat” and “medv”.
Standardized data
boston_scaled<-scale(Boston)
summary(boston_scaled)## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865Covert matrix data to dafaframe
class(boston_scaled)## [1] "matrix"boston_scaled<-as.data.frame(boston_scaled)Create a catergorical variable with the variable “Crime”
summary("crim")## Length Class Mode
## 1 character characterbins<-quantile(boston_scaled$crim)
bins## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610crime<-cut(boston_scaled$crim, breaks=bins, include.lowest=TRUE, label=c("low","med_low","med_high", "high"))
crime## [1] low low low low low low med_low
## [8] med_low med_low med_low med_low med_low med_low med_high
## [15] med_high med_high med_high med_high med_high med_high med_high
## [22] med_high med_high med_high med_high med_high med_high med_high
## [29] med_high med_high med_high med_high med_high med_high med_high
## [36] low med_low low med_low low low med_low
## [43] med_low med_low med_low med_low med_low med_low med_low
## [50] med_low med_low low low low low low
## [57] low low med_low med_low med_low med_low med_low
## [64] med_low low low low low med_low med_low
## [71] med_low med_low med_low med_low low med_low med_low
## [78] med_low low med_low low low low low
## [85] low low low low low low low
## [92] low low low low med_low med_low med_low
## [99] low low med_low med_low med_low med_low med_low
## [106] med_low med_low med_low med_low med_high med_low med_low
## [113] med_low med_low med_low med_low med_low med_low med_low
## [120] med_low low low med_low med_low med_low med_low
## [127] med_high med_high med_high med_high med_high med_high med_high
## [134] med_high med_high med_high med_high med_high med_low med_high
## [141] med_high med_high med_high high med_high med_high med_high
## [148] med_high med_high med_high med_high med_high med_high med_high
## [155] med_high med_high med_high med_high med_high med_high med_high
## [162] med_high med_high med_high med_high med_high med_high med_high
## [169] med_high med_high med_high med_high med_low med_low med_low
## [176] low low low low low low low
## [183] med_low med_low med_low low low low med_low
## [190] med_low med_low low med_low low low low
## [197] low low low low low low low
## [204] low low med_low med_low med_low med_low med_high
## [211] med_low med_high med_low med_low med_high med_low low
## [218] low med_low med_low med_high med_high med_high med_high
## [225] med_high med_high med_high med_high med_high med_high med_high
## [232] med_high med_high med_high med_high med_high med_high med_high
## [239] med_low med_low med_low med_low med_low med_low med_low
## [246] med_low med_high med_low med_low med_low med_low med_low
## [253] med_low med_high low low low med_high med_high
## [260] med_high med_high med_high med_high med_high med_high med_high
## [267] med_high med_high med_high med_low med_high med_low med_low
## [274] med_low low med_low med_low low low med_low
## [281] low low low low low low low
## [288] low low low low low low med_low
## [295] low med_low low med_low low low low
## [302] low med_low med_low low low low low
## [309] med_high med_high med_high med_high med_high med_high med_high
## [316] med_low med_high med_low med_high med_high med_low med_low
## [323] med_high med_high med_high med_low med_high med_low low
## [330] low low low low low low low
## [337] low low low low low low low
## [344] low low low low low low low
## [351] low low low low low med_low high
## [358] high high high high high high high
## [365] med_high high high high high high high
## [372] high high high high high high high
## [379] high high high high high high high
## [386] high high high high high high high
## [393] high high high high high high high
## [400] high high high high high high high
## [407] high high high high high high high
## [414] high high high high high high high
## [421] high high high high high high high
## [428] high high high high high high high
## [435] high high high high high high high
## [442] high high high high high high high
## [449] high high high high high high high
## [456] high high high high high high high
## [463] high high high med_high high high high
## [470] high high high med_high high high high
## [477] high high high high high high high
## [484] med_high med_high med_high high high med_low med_low
## [491] med_low med_low med_low med_low med_high med_low med_high
## [498] med_high med_low med_low med_low low low low
## [505] med_low low
## Levels: low med_low med_high hightable(crime)## crime
## low med_low med_high high
## 127 126 126 127Remove original “crim” from the dataset and add the new categorial variable
boston_scaled<-dplyr::select(boston_scaled,-crim)
boston_scaled## zn indus chas nox rm
## 1 0.28454827 -1.28663623 -0.2723291 -0.14407485 0.413262920
## 2 -0.48724019 -0.59279438 -0.2723291 -0.73953036 0.194082387
## 3 -0.48724019 -0.59279438 -0.2723291 -0.73953036 1.281445551
## 4 -0.48724019 -1.30558569 -0.2723291 -0.83445805 1.015297761
## 5 -0.48724019 -1.30558569 -0.2723291 -0.83445805 1.227362043
## 6 -0.48724019 -1.30558569 -0.2723291 -0.83445805 0.206891639
## 7 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.388026950
## 8 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.160306916
## 9 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.930285282
## 10 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.399412952
## 11 0.04872402 -0.47618230 -0.2723291 -0.26489191 0.131459378
## 12 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.392296701
## 13 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.563086727
## 14 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.477691714
## 15 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 16 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.641365488
## 17 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.497617217
## 18 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.419338455
## 19 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.179354069
## 20 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.793653261
## 21 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.017103545
## 22 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.454919710
## 23 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.203004422
## 24 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.671253743
## 25 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.513272969
## 26 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.975829289
## 27 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.671253743
## 28 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.338213193
## 29 -0.48724019 -0.43682573 -0.2723291 -0.14407485 0.299402903
## 30 -0.48724019 -0.43682573 -0.2723291 -0.14407485 0.554164692
## 31 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.813578764
## 32 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.302631937
## 33 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.476268464
## 34 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.830657767
## 35 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 36 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.500463717
## 37 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.631402737
## 38 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.618593485
## 39 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.453496460
## 40 2.72854505 -1.19334657 -0.2723291 -1.09335175 0.441727925
## 41 2.72854505 -1.19334657 -0.2723291 -1.09335175 1.052302267
## 42 -0.48724019 -0.61611679 -0.2723291 -0.92075595 0.690796712
## 43 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.164576667
## 44 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.104800158
## 45 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.306901688
## 46 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.857699521
## 47 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.709681499
## 48 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.362408446
## 49 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -1.260479332
## 50 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.971559538
## 51 0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.457766211
## 52 0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.241432178
## 53 0.41317968 -0.80123846 -0.2723291 -0.99842406 0.322174907
## 54 0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.407952453
## 55 2.72854505 -1.04029322 -0.2723291 -1.24868797 -0.564509977
## 56 3.37170210 -1.44552019 -0.2723291 -1.30909650 1.372533565
## 57 3.15731641 -1.51548743 -0.2723291 -1.24868797 0.139998879
## 58 3.80047346 -1.43094368 -0.2723291 -1.24005818 0.756266222
## 59 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.198734672
## 60 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.509003218
## 61 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.773727758
## 62 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.453496460
## 63 0.58468822 -0.87557866 -0.2723291 -0.87760700 0.243896145
## 64 0.58468822 -0.87557866 -0.2723291 -0.87760700 0.679410711
## 65 0.26310970 -1.42219777 -0.2723291 -1.19604625 1.166162284
## 66 2.94293073 -1.13212523 -0.2723291 -1.35224545 0.007636609
## 67 2.94293073 -1.13212523 -0.2723291 -1.35224545 -0.708258248
## 68 0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.578742479
## 69 0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.982945540
## 70 0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.568779727
## 71 -0.48724019 -0.04763292 -0.2723291 -1.22279860 0.188389387
## 72 -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.460612711
## 73 -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.312594689
## 74 -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.056409650
## 75 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.016558644
## 76 -0.48724019 0.24681257 -0.2723291 -1.01568364 0.001943608
## 77 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.008019143
## 78 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.205850923
## 79 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.074911903
## 80 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.584435480
## 81 0.58468822 -0.91493524 -0.2723291 -1.11061133 0.629596953
## 82 0.58468822 -0.91493524 -0.2723291 -1.11061133 0.475885930
## 83 0.58468822 -0.91493524 -0.2723291 -1.11061133 0.024715612
## 84 0.58468822 -0.91493524 -0.2723291 -1.11061133 -0.167423167
## 85 -0.48724019 -0.96886832 -0.2723291 -0.91212616 0.148538381
## 86 -0.48724019 -0.96886832 -0.2723291 -0.91212616 0.491541682
## 87 -0.48724019 -0.96886832 -0.2723291 -0.91212616 -0.383757200
## 88 -0.48724019 -0.96886832 -0.2723291 -0.91212616 -0.232892677
## 89 -0.48724019 -1.12629463 -0.2723291 -0.56693456 1.028107013
## 90 -0.48724019 -1.12629463 -0.2723291 -0.56693456 1.130581029
## 91 -0.48724019 -1.12629463 -0.2723291 -0.56693456 0.188389387
## 92 -0.48724019 -1.12629463 -0.2723291 -0.56693456 0.171310384
## 93 0.71331963 0.56895343 -0.2723291 -0.78267931 0.223970642
## 94 0.71331963 0.56895343 -0.2723291 -0.78267931 -0.104800158
## 95 0.71331963 0.56895343 -0.2723291 -0.78267931 -0.050716649
## 96 -0.48724019 -1.20209248 -0.2723291 -0.94664532 0.484425431
## 97 -0.48724019 -1.20209248 -0.2723291 -0.94664532 -0.173116168
## 98 -0.48724019 -1.20209248 -0.2723291 -0.94664532 2.539598741
## 99 -0.48724019 -1.20209248 -0.2723291 -0.94664532 2.185209437
## 100 -0.48724019 -1.20209248 -0.2723291 -0.94664532 1.610216351
## 101 -0.48724019 -0.37560439 -0.2723291 -0.29941107 0.629596953
## 102 -0.48724019 -0.37560439 -0.2723291 -0.29941107 0.706452465
## 103 -0.48724019 -0.37560439 -0.2723291 -0.29941107 0.171310384
## 104 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.210120673
## 105 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.167423167
## 106 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.617170235
## 107 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.638518988
## 108 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.224353176
## 109 -0.48724019 -0.37560439 -0.2723291 -0.29941107 0.269514649
## 110 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.079181654
## 111 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.127572161
## 112 -0.48724019 -0.16424500 -0.2723291 -0.06640675 0.612517950
## 113 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.528928721
## 114 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.274166933
## 115 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.043600398
## 116 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.507579968
## 117 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.154613915
## 118 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.375217698
## 119 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.587281980
## 120 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.787960260
## 121 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.590128481
## 122 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.399412952
## 123 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.460612711
## 124 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.610053984
## 125 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.577319229
## 126 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.425031456
## 127 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.955903786
## 128 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.842043769
## 129 -0.48724019 1.56744433 -0.2723291 0.59808708 0.208314890
## 130 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.921745781
## 131 -0.48724019 1.56744433 -0.2723291 0.59808708 0.246742645
## 132 -0.48724019 1.56744433 -0.2723291 0.59808708 0.058873617
## 133 -0.48724019 1.56744433 -0.2723291 0.59808708 0.124343127
## 134 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.658444491
## 135 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.750955755
## 136 -0.48724019 1.56744433 -0.2723291 0.59808708 0.071682869
## 137 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.487654465
## 138 -0.48724019 1.56744433 -0.2723291 0.59808708 0.241049645
## 139 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.608630733
## 140 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.190195170
## 141 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.157460416
## 142 -0.48724019 1.56744433 -0.2723291 0.59808708 -1.801314413
## 143 -0.48724019 1.23072696 3.6647712 2.72964520 -1.254786331
## 144 -0.48724019 1.23072696 -0.2723291 2.72964520 -1.162275067
## 145 -0.48724019 1.23072696 -0.2723291 2.72964520 -1.966411438
## 146 -0.48724019 1.23072696 -0.2723291 2.72964520 -0.220083425
## 147 -0.48724019 1.23072696 -0.2723291 2.72964520 -0.934555033
## 148 -0.48724019 1.23072696 -0.2723291 2.72964520 -1.933676683
## 149 -0.48724019 1.23072696 -0.2723291 2.72964520 -1.563631627
## 150 -0.48724019 1.23072696 -0.2723291 2.72964520 -0.978675789
## 151 -0.48724019 1.23072696 -0.2723291 2.72964520 -0.231469427
## 152 -0.48724019 1.23072696 -0.2723291 2.72964520 -1.253363081
## 153 -0.48724019 1.23072696 3.6647712 2.72964520 -1.811277165
## 154 -0.48724019 1.23072696 -0.2723291 2.72964520 -0.819271765
## 155 -0.48724019 1.23072696 3.6647712 2.72964520 -0.221506675
## 156 -0.48724019 1.23072696 3.6647712 2.72964520 -0.188771920
## 157 -0.48724019 1.23072696 -0.2723291 2.72964520 -1.441232109
## 158 -0.48724019 1.23072696 -0.2723291 0.43412107 0.937018999
## 159 -0.48724019 1.23072696 -0.2723291 0.43412107 -0.311171439
## 160 -0.48724019 1.23072696 -0.2723291 2.72964520 0.320751657
## 161 -0.48724019 1.23072696 3.6647712 0.43412107 -0.049293399
## 162 -0.48724019 1.23072696 -0.2723291 0.43412107 1.714113616
## 163 -0.48724019 1.23072696 3.6647712 0.43412107 2.159590934
## 164 -0.48724019 1.23072696 3.6647712 0.43412107 2.975113306
## 165 -0.48724019 1.23072696 -0.2723291 0.43412107 -0.612900484
## 166 -0.48724019 1.23072696 -0.2723291 0.43412107 -0.261357681
## 167 -0.48724019 1.23072696 -0.2723291 0.43412107 2.340343711
## 168 -0.48724019 1.23072696 -0.2723291 0.43412107 -0.580165729
## 169 -0.48724019 1.23072696 -0.2723291 0.43412107 0.048910866
## 170 -0.48724019 1.23072696 -0.2723291 0.43412107 0.167040633
## 171 -0.48724019 1.23072696 -0.2723291 0.43412107 -0.583012230
## 172 -0.48724019 1.23072696 -0.2723291 0.43412107 -0.575895979
## 173 -0.48724019 -1.03300497 -0.2723291 -0.38570897 -1.014257045
## 174 -0.48724019 -1.03300497 -0.2723291 -0.38570897 0.186966136
## 175 -0.48724019 -1.03300497 -0.2723291 -0.38570897 -0.605784233
## 176 -0.48724019 -1.03300497 -0.2723291 -0.38570897 0.371988664
## 177 -0.48724019 -1.03300497 -0.2723291 -0.38570897 -0.376640949
## 178 -0.48724019 -1.03300497 -0.2723291 -0.38570897 0.043217865
## 179 -0.48724019 -1.03300497 -0.2723291 -0.38570897 0.818889232
## 180 -0.48724019 -1.26477147 -0.2723291 -0.57556435 0.989679257
## 181 -0.48724019 -1.26477147 -0.2723291 -0.57556435 2.106930676
## 182 -0.48724019 -1.26477147 -0.2723291 -0.57556435 -0.200157922
## 183 -0.48724019 -1.26477147 -0.2723291 -0.57556435 1.238748045
## 184 -0.48724019 -1.26477147 -0.2723291 -0.57556435 0.396183918
## 185 -0.48724019 -1.26477147 -0.2723291 -0.57556435 -0.968713038
## 186 -0.48724019 -1.26477147 -0.2723291 -0.57556435 -0.187348670
## 187 -0.48724019 -1.26477147 -0.2723291 -0.57556435 2.200865190
## 188 1.44223095 -1.12192167 -0.2723291 -1.01568364 0.707875715
## 189 1.44223095 -1.12192167 -0.2723291 -1.01568364 0.386221166
## 190 1.44223095 -1.12192167 -0.2723291 -1.01568364 1.281445551
## 191 1.44223095 -1.12192167 -0.2723291 -1.01568364 0.948405001
## 192 1.44223095 -1.12192167 -0.2723291 -1.01568364 0.646675956
## 193 1.44223095 -1.12192167 -0.2723291 -1.01568364 1.271482800
## 194 2.08538800 -1.19626187 -0.2723291 -1.32635608 0.733494219
## 195 2.08538800 -1.19626187 -0.2723291 -1.32635608 0.454537177
## 196 2.94293073 -1.55630166 -0.2723291 -1.14513049 2.263488199
## 197 2.94293073 -1.40179066 -0.2723291 -1.30046671 1.426617073
## 198 2.94293073 -1.40179066 -0.2723291 -1.30046671 1.170432035
## 199 2.94293073 -1.40179066 -0.2723291 -1.30046671 1.408114820
## 200 3.58608778 -1.40907891 -0.2723291 -1.30909650 0.982563006
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## 436 -0.48724019 1.01499462 -0.2723291 1.59914271 0.490118432
## 437 -0.48724019 1.01499462 -0.2723291 1.59914271 0.251012396
## 438 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.188771920
## 439 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.497617217
## 440 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.935978283
## 441 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.664137492
## 442 -0.48724019 1.01499462 -0.2723291 1.59914271 0.172733634
## 443 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.093414156
## 444 -0.48724019 1.01499462 -0.2723291 1.59914271 0.285170401
## 445 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.612900484
## 446 -0.48724019 1.01499462 -0.2723291 1.59914271 0.248165896
## 447 -0.48724019 1.01499462 -0.2723291 1.59914271 0.080222370
## 448 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.047870149
## 449 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.141804663
## 450 -0.48724019 1.01499462 -0.2723291 1.36613839 0.188389387
## 451 -0.48724019 1.01499462 -0.2723291 1.36613839 0.660908458
## 452 -0.48724019 1.01499462 -0.2723291 1.36613839 0.527122938
## 453 -0.48724019 1.01499462 -0.2723291 1.36613839 0.017599361
## 454 -0.48724019 1.01499462 -0.2723291 1.36613839 1.577481596
## 455 -0.48724019 1.01499462 -0.2723291 1.36613839 0.631020203
## 456 -0.48724019 1.01499462 -0.2723291 1.36613839 0.342100410
## 457 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.439263958
## 458 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.496193967
## 459 -0.48724019 1.01499462 -0.2723291 1.36613839 0.023292362
## 460 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.289822685
## 461 -0.48724019 1.01499462 -0.2723291 1.36613839 0.592592447
## 462 -0.48724019 1.01499462 -0.2723291 1.36613839 0.130036128
## 463 -0.48724019 1.01499462 -0.2723291 1.36613839 0.046064365
## 464 -0.48724019 1.01499462 -0.2723291 1.36613839 0.325021407
## 465 -0.48724019 1.01499462 -0.2723291 0.86561057 -0.107646658
## 466 -0.48724019 1.01499462 -0.2723291 0.86561057 -0.748109254
## 467 -0.48724019 1.01499462 -0.2723291 0.86561057 -0.473421963
## 468 -0.48724019 1.01499462 -0.2723291 0.25289548 -0.400836202
## 469 -0.48724019 1.01499462 -0.2723291 0.21837632 -0.510426469
## 470 -0.48724019 1.01499462 -0.2723291 0.21837632 -0.813578764
## 471 -0.48724019 1.01499462 -0.2723291 0.21837632 -0.167423167
## 472 -0.48724019 1.01499462 -0.2723291 -0.19585359 -0.079181654
## 473 -0.48724019 1.01499462 -0.2723291 0.21837632 0.216854391
## 474 -0.48724019 1.01499462 -0.2723291 0.51178918 0.989679257
## 475 -0.48724019 1.01499462 -0.2723291 0.25289548 -1.220628326
## 476 -0.48724019 1.01499462 -0.2723291 0.25289548 -0.174539418
## 477 -0.48724019 1.01499462 -0.2723291 0.51178918 0.283747151
## 478 -0.48724019 1.01499462 -0.2723291 0.51178918 -1.395688102
## 479 -0.48724019 1.01499462 -0.2723291 0.51178918 -0.141804663
## 480 -0.48724019 1.01499462 -0.2723291 0.51178918 -0.079181654
## 481 -0.48724019 1.01499462 -0.2723291 -0.19585359 -0.060679401
## 482 -0.48724019 1.01499462 -0.2723291 -0.19585359 0.662331708
## 483 -0.48724019 1.01499462 -0.2723291 -0.19585359 1.104962525
## 484 -0.48724019 1.01499462 -0.2723291 -0.19585359 -0.743839504
## 485 -0.48724019 1.01499462 -0.2723291 0.24426569 -0.588705230
## 486 -0.48724019 1.01499462 -0.2723291 0.24426569 0.038948114
## 487 -0.48724019 1.01499462 -0.2723291 0.24426569 -0.242855428
## 488 -0.48724019 1.01499462 -0.2723291 0.24426569 -0.540314723
## 489 -0.48724019 2.42017014 -0.2723291 0.46864023 -1.182200570
## 490 -0.48724019 2.42017014 -0.2723291 0.46864023 -1.239130578
## 491 -0.48724019 2.42017014 -0.2723291 0.46864023 -1.695993897
## 492 -0.48724019 2.42017014 -0.2723291 0.46864023 -0.429301206
## 493 -0.48724019 2.42017014 -0.2723291 0.46864023 -0.429301206
## 494 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.822118266
## 495 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.510426469
## 496 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.874778524
## 497 -0.48724019 -0.21088983 -0.2723291 0.26152527 -1.273288584
## 498 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.698295497
## 499 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.378064199
## 500 -0.48724019 -0.21088983 -0.2723291 0.26152527 -1.018526795
## 501 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.366678197
## 502 -0.48724019 0.11562398 -0.2723291 0.15796779 0.438881424
## 503 -0.48724019 0.11562398 -0.2723291 0.15796779 -0.234315927
## 504 -0.48724019 0.11562398 -0.2723291 0.15796779 0.983986256
## 505 -0.48724019 0.11562398 -0.2723291 0.15796779 0.724954717
## 506 -0.48724019 0.11562398 -0.2723291 0.15796779 -0.362408446
## age dis rad tax ptratio
## 1 -0.119894767 0.1400749840 -0.9818712 -0.66594918 -1.45755797
## 2 0.366803426 0.5566090496 -0.8670245 -0.98635338 -0.30279450
## 3 -0.265548971 0.5566090496 -0.8670245 -0.98635338 -0.30279450
## 4 -0.809087830 1.0766711351 -0.7521778 -1.10502160 0.11292035
## 5 -0.510674339 1.0766711351 -0.7521778 -1.10502160 0.11292035
## 6 -0.350809969 1.0766711351 -0.7521778 -1.10502160 0.11292035
## 7 -0.070159185 0.8384142195 -0.5224844 -0.57694801 -1.50374851
## 8 0.977840575 1.0236248974 -0.5224844 -0.57694801 -1.50374851
## 9 1.116389695 1.0861216287 -0.5224844 -0.57694801 -1.50374851
## 10 0.615481336 1.3283202075 -0.5224844 -0.57694801 -1.50374851
## 11 0.913894827 1.2117799501 -0.5224844 -0.57694801 -1.50374851
## 12 0.508905089 1.1547920492 -0.5224844 -0.57694801 -1.50374851
## 13 -1.050660656 0.7863652700 -0.5224844 -0.57694801 -1.50374851
## 14 -0.240681180 0.4333252240 -0.6373311 -0.60068166 1.17530274
## 15 0.565745754 0.3166899868 -0.6373311 -0.60068166 1.17530274
## 16 -0.428965883 0.3341187865 -0.6373311 -0.60068166 1.17530274
## 17 -1.395257187 0.3341187865 -0.6373311 -0.60068166 1.17530274
## 18 0.466274590 0.2198105553 -0.6373311 -0.60068166 1.17530274
## 19 -1.135921653 0.0006920764 -0.6373311 -0.60068166 1.17530274
## 20 0.032864520 0.0006920764 -0.6373311 -0.60068166 1.17530274
## 21 1.048891406 0.0013569352 -0.6373311 -0.60068166 1.17530274
## 22 0.732715207 0.1031753182 -0.6373311 -0.60068166 1.17530274
## 23 0.821528746 0.0863638874 -0.6373311 -0.60068166 1.17530274
## 24 1.116389695 0.1425444597 -0.6373311 -0.60068166 1.17530274
## 25 0.906789743 0.2871037683 -0.6373311 -0.60068166 1.17530274
## 26 0.608376252 0.3132232229 -0.6373311 -0.60068166 1.17530274
## 27 0.771793164 0.4212152950 -0.6373311 -0.60068166 1.17530274
## 28 0.718505041 0.3126533438 -0.6373311 -0.60068166 1.17530274
## 29 0.917447368 0.3132707128 -0.6373311 -0.60068166 1.17530274
## 30 0.665216917 0.2108349609 -0.6373311 -0.60068166 1.17530274
## 31 0.906789743 0.2079855659 -0.6373311 -0.60068166 1.17530274
## 32 1.116389695 0.1804414138 -0.6373311 -0.60068166 1.17530274
## 33 0.476932215 0.0925850666 -0.6373311 -0.60068166 1.17530274
## 34 0.938762618 -0.0037244859 -0.6373311 -0.60068166 1.17530274
## 35 1.006260907 -0.0167367233 -0.6373311 -0.60068166 1.17530274
## 36 -0.013318520 -0.2064589434 -0.5224844 -0.76681717 0.34387304
## 37 -0.254891346 -0.1981007179 -0.5224844 -0.76681717 0.34387304
## 38 -0.961847117 0.0660856927 -0.5224844 -0.76681717 0.34387304
## 39 -1.363284313 0.0248169544 -0.5224844 -0.76681717 0.34387304
## 40 -1.661697804 0.7627152911 -0.7521778 -0.92701927 -0.07184181
## 41 -1.874850298 0.7627152911 -0.7521778 -0.92701927 -0.07184181
## 42 -2.333128159 0.9145880470 -0.7521778 -1.03975408 -0.25660396
## 43 -2.201684121 0.9145880470 -0.7521778 -1.03975408 -0.25660396
## 44 -2.205236663 0.9145880470 -0.7521778 -1.03975408 -0.25660396
## 45 -1.015135240 0.9145880470 -0.7521778 -1.03975408 -0.25660396
## 46 -1.235392817 0.6199131095 -0.7521778 -1.03975408 -0.25660396
## 47 -1.253155525 0.6199131095 -0.7521778 -1.03975408 -0.25660396
## 48 0.601271169 0.8996287230 -0.7521778 -1.03975408 -0.25660396
## 49 0.949420242 0.9853955139 -0.7521778 -1.03975408 -0.25660396
## 50 -0.233576097 1.0887810641 -0.7521778 -1.03975408 -0.25660396
## 51 -0.812640371 1.4340327636 -0.6373311 -0.98041997 -0.76469989
## 52 -0.198050682 1.4340327636 -0.6373311 -0.98041997 -0.76469989
## 53 -1.686565595 1.4340327636 -0.6373311 -0.98041997 -0.76469989
## 54 -1.675907970 1.4340327636 -0.6373311 -0.98041997 -0.76469989
## 55 -0.745142082 1.6738568465 -0.7521778 0.36053095 1.22149328
## 56 -1.658145262 2.3277455193 -0.5224844 -1.08128796 -0.25660396
## 57 -1.167894527 2.5609210138 -0.8670245 -0.56508119 -0.53374719
## 58 -0.997372532 2.1511780064 -0.5224844 -0.90328562 -1.54993904
## 59 -1.398809729 1.9089794276 -0.1779443 -0.73715011 0.57482574
## 60 -0.759352248 1.4897384367 -0.1779443 -0.73715011 0.57482574
## 61 -0.084369352 1.6290738544 -0.1779443 -0.73715011 0.57482574
## 62 0.881921953 1.4358373805 -0.1779443 -0.73715011 0.57482574
## 63 -0.027528687 1.6291213443 -0.1779443 -0.73715011 0.57482574
## 64 -0.894348827 1.9878601804 -0.1779443 -0.73715011 0.57482574
## 65 -0.322389636 2.5776849546 -0.7521778 -1.14062207 0.06672981
## 66 -1.803799466 1.3375332514 -0.6373311 -0.42267932 -1.08803366
## 67 -1.331311439 1.3375332514 -0.6373311 -0.42267932 -1.08803366
## 68 -1.675907970 1.2836321952 -0.6373311 -0.37521203 0.20530143
## 69 -1.128816570 1.2836321952 -0.6373311 -0.37521203 0.20530143
## 70 -1.263813149 1.2836321952 -0.6373311 -0.37521203 0.20530143
## 71 -2.201684121 0.7086717651 -0.6373311 -0.61254848 0.34387304
## 72 -1.814457091 0.7086717651 -0.6373311 -0.61254848 0.34387304
## 73 -2.159053622 0.7086717651 -0.6373311 -0.61254848 0.34387304
## 74 -2.215894287 0.7086717651 -0.6373311 -0.61254848 0.34387304
## 75 -2.222999370 0.2167712006 -0.5224844 -0.06074124 0.11292035
## 76 -0.837508162 0.3360183832 -0.5224844 -0.06074124 0.11292035
## 77 0.210491598 0.1221237952 -0.5224844 -0.06074124 0.11292035
## 78 -0.809087830 0.1403124336 -0.5224844 -0.06074124 0.11292035
## 79 -0.528437047 0.5789293108 -0.5224844 -0.06074124 0.11292035
## 80 -1.135921653 0.3360183832 -0.5224844 -0.06074124 0.11292035
## 81 -1.246050442 0.7625253315 -0.6373311 -0.75495035 0.25149196
## 82 0.064837394 0.7625253315 -0.6373311 -0.75495035 0.25149196
## 83 -1.292233482 0.7625253315 -0.6373311 -0.75495035 0.25149196
## 84 -0.777114956 0.7625253315 -0.6373311 -0.75495035 0.25149196
## 85 -0.730931915 0.4674704746 -0.7521778 -0.95668632 0.02053927
## 86 -0.443176049 0.3051974268 -0.7521778 -0.95668632 0.02053927
## 87 -0.833955621 0.3002109855 -0.7521778 -0.95668632 0.02053927
## 88 -0.418308258 -0.0225304932 -0.7521778 -0.95668632 0.02053927
## 89 0.629691502 -0.1773001341 -0.8670245 -0.82021787 -0.30279450
## 90 -0.194498140 -0.1807194081 -0.8670245 -0.82021787 -0.30279450
## 91 -0.087921893 -0.3337319220 -0.8670245 -0.82021787 -0.30279450
## 92 0.189176348 -0.3338269018 -0.8670245 -0.82021787 -0.30279450
## 93 -0.531989588 -0.0613297558 -0.6373311 -0.82021787 -0.11803234
## 94 -1.409467353 -0.0613297558 -0.6373311 -0.82021787 -0.11803234
## 95 0.309962761 -0.0855021237 -0.6373311 -0.82021787 -0.11803234
## 96 -0.382782843 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 97 0.036417061 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 98 0.263779721 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 99 -1.125264029 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 100 -0.215813389 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 101 0.402328842 -0.4830877123 -0.5224844 -0.14380900 1.12911220
## 102 0.096810268 -0.4459031069 -0.5224844 -0.14380900 1.12911220
## 103 0.597718628 -0.5130538501 -0.5224844 -0.14380900 1.12911220
## 104 0.668769459 -0.5130538501 -0.5224844 -0.14380900 1.12911220
## 105 0.761135540 -0.6525317376 -0.5224844 -0.14380900 1.12911220
## 106 0.999155824 -0.8016975682 -0.5224844 -0.14380900 1.12911220
## 107 0.828633829 -0.7522605641 -0.5224844 -0.14380900 1.12911220
## 108 0.590613545 -0.7943366310 -0.5224844 -0.14380900 1.12911220
## 109 1.013365990 -0.6468804374 -0.5224844 -0.14380900 1.12911220
## 110 0.803766038 -0.5935967501 -0.5224844 -0.14380900 1.12911220
## 111 -0.503569256 -0.4830877123 -0.5224844 -0.14380900 1.12911220
## 112 0.462722049 -0.5307200994 -0.4076377 0.14099473 -0.30279450
## 113 0.864159245 -0.6846349217 -0.4076377 0.14099473 -0.30279450
## 114 0.952972784 -0.5922195425 -0.4076377 0.14099473 -0.30279450
## 115 0.555088129 -0.7306526517 -0.4076377 0.14099473 -0.30279450
## 116 0.697189791 -0.6325384823 -0.4076377 0.14099473 -0.30279450
## 117 0.139440767 -0.5057404029 -0.4076377 0.14099473 -0.30279450
## 118 0.498247464 -0.4975246471 -0.4076377 0.14099473 -0.30279450
## 119 0.160756016 -0.6256999342 -0.4076377 0.14099473 -0.30279450
## 120 -0.119894767 -0.4919208369 -0.4076377 0.14099473 -0.30279450
## 121 0.039969603 -0.7300827727 -0.8670245 -1.30675758 0.29768250
## 122 0.551535588 -0.7587191929 -0.8670245 -1.30675758 0.29768250
## 123 0.864159245 -0.8111955516 -0.8670245 -1.30675758 0.29768250
## 124 1.009813449 -0.8788686839 -0.8670245 -1.30675758 0.29768250
## 125 0.967182950 -0.8494724251 -0.8670245 -1.30675758 0.29768250
## 126 0.704294875 -0.8558360740 -0.8670245 -1.30675758 0.29768250
## 127 0.960077867 -0.9677698093 -0.8670245 -1.30675758 0.29768250
## 128 0.974288033 -0.9530004450 -0.6373311 0.17066179 1.26768382
## 129 1.073759197 -0.9415078850 -0.6373311 0.17066179 1.26768382
## 130 0.928104993 -0.8620097633 -0.6373311 0.17066179 1.26768382
## 131 1.077311738 -0.7961887377 -0.6373311 0.17066179 1.26768382
## 132 1.034681240 -0.7237666137 -0.6373311 0.17066179 1.26768382
## 133 1.041786323 -0.6969823003 -0.6373311 0.17066179 1.26768382
## 134 0.952972784 -0.6293091680 -0.6373311 0.17066179 1.26768382
## 135 1.059549031 -0.6881491756 -0.6373311 0.17066179 1.26768382
## 136 1.052443947 -0.7998929513 -0.6373311 0.17066179 1.26768382
## 137 0.885474494 -0.8681834525 -0.6373311 0.17066179 1.26768382
## 138 1.059549031 -0.9237941458 -0.6373311 0.17066179 1.26768382
## 139 1.052443947 -1.0098458762 -0.6373311 0.17066179 1.26768382
## 140 1.041786323 -1.0097983862 -0.6373311 0.17066179 1.26768382
## 141 0.889027036 -1.0367726593 -0.6373311 0.17066179 1.26768382
## 142 1.116389695 -1.1186927669 -0.6373311 0.17066179 1.26768382
## 143 1.116389695 -1.1746358896 -0.5224844 -0.03107419 -1.73470120
## 144 1.116389695 -1.1317999841 -0.5224844 -0.03107419 -1.73470120
## 145 1.038233781 -1.1630958396 -0.5224844 -0.03107419 -1.73470120
## 146 1.116389695 -1.1283332201 -0.5224844 -0.03107419 -1.73470120
## 147 1.116389695 -1.0820305506 -0.5224844 -0.03107419 -1.73470120
## 148 0.963630408 -1.1085299245 -0.5224844 -0.03107419 -1.73470120
## 149 0.896132119 -1.0758568614 -0.5224844 -0.03107419 -1.73470120
## 150 0.935210076 -1.0777089681 -0.5224844 -0.03107419 -1.73470120
## 151 1.020471073 -1.0338757744 -0.5224844 -0.03107419 -1.73470120
## 152 1.116389695 -1.0464131126 -0.5224844 -0.03107419 -1.73470120
## 153 0.690084708 -1.0375799879 -0.5224844 -0.03107419 -1.73470120
## 154 1.063101572 -1.0314062987 -0.5224844 -0.03107419 -1.73470120
## 155 0.974288033 -0.9714740229 -0.5224844 -0.03107419 -1.73470120
## 156 0.498247464 -0.9733261297 -0.5224844 -0.03107419 -1.73470120
## 157 0.903237202 -0.9776477121 -0.5224844 -0.03107419 -1.73470120
## 158 1.024023615 -0.9107344185 -0.5224844 -0.03107419 -1.73470120
## 159 1.116389695 -0.9677223194 -0.5224844 -0.03107419 -1.73470120
## 160 1.116389695 -0.9636381865 -0.5224844 -0.03107419 -1.73470120
## 161 0.853501620 -0.9482039634 -0.5224844 -0.03107419 -1.73470120
## 162 0.789555872 -0.8662838558 -0.5224844 -0.03107419 -1.73470120
## 163 1.052443947 -0.8331358935 -0.5224844 -0.03107419 -1.73470120
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## 393 1.009813449 -0.9616910999 1.6596029 1.52941294 0.80577843
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## 395 0.928104993 -0.9559448199 1.6596029 1.52941294 0.80577843
## 396 1.073759197 -0.9827291333 1.6596029 1.52941294 0.80577843
## 397 0.974288033 -1.0059517029 1.6596029 1.52941294 0.80577843
## 398 1.077311738 -1.0265623271 1.6596029 1.52941294 0.80577843
## 399 1.116389695 -1.0948528283 1.6596029 1.52941294 0.80577843
## 400 0.327725469 -1.0897239172 1.6596029 1.52941294 0.80577843
## 401 1.116389695 -1.0477428302 1.6596029 1.52941294 0.80577843
## 402 1.116389695 -1.0547238481 1.6596029 1.52941294 0.80577843
## 403 1.116389695 -1.0239028917 1.6596029 1.52941294 0.80577843
## 404 0.974288033 -0.9936043244 1.6596029 1.52941294 0.80577843
## 405 0.597718628 -1.0389097056 1.6596029 1.52941294 0.80577843
## 406 1.116389695 -1.1253413553 1.6596029 1.52941294 0.80577843
## 407 1.116389695 -1.2427839210 1.6596029 1.52941294 0.80577843
## 408 1.116389695 -1.1919222195 1.6596029 1.52941294 0.80577843
## 409 1.041786323 -1.1114268095 1.6596029 1.52941294 0.80577843
## 410 1.116389695 -1.1062978984 1.6596029 1.52941294 0.80577843
## 411 1.116389695 -1.1312301050 1.6596029 1.52941294 0.80577843
## 412 1.116389695 -1.0768541496 1.6596029 1.52941294 0.80577843
## 413 1.116389695 -1.0643168114 1.6596029 1.52941294 0.80577843
## 414 1.116389695 -1.0474578907 1.6596029 1.52941294 0.80577843
## 415 1.116389695 -1.0147848276 1.6596029 1.52941294 0.80577843
## 416 1.116389695 -0.9309651233 1.6596029 1.52941294 0.80577843
## 417 0.789555872 -0.9381835908 1.6596029 1.52941294 0.80577843
## 418 0.729162665 -1.0198662487 1.6596029 1.52941294 0.80577843
## 419 1.116389695 -0.9462093868 1.6596029 1.52941294 0.80577843
## 420 0.281542429 -0.9502935197 1.6596029 1.52941294 0.80577843
## 421 1.116389695 -0.9194725633 1.6596029 1.52941294 0.80577843
## 422 0.949420242 -0.9120166463 1.6596029 1.52941294 0.80577843
## 423 0.675874542 -0.8756393696 1.6596029 1.52941294 0.80577843
## 424 0.587061003 -0.8421114879 1.6596029 1.52941294 0.80577843
## 425 0.071942477 -0.8223081923 1.6596029 1.52941294 0.80577843
## 426 0.952972784 -0.8953951752 1.6596029 1.52941294 0.80577843
## 427 -0.315284553 -0.8536040479 1.6596029 1.52941294 0.80577843
## 428 0.359698343 -0.9175729666 1.6596029 1.52941294 0.80577843
## 429 0.338383094 -0.8830477967 1.6596029 1.52941294 0.80577843
## 430 0.960077867 -0.8675660836 1.6596029 1.52941294 0.80577843
## 431 0.622586419 -0.8274371034 1.6596029 1.52941294 0.80577843
## 432 0.913894827 -0.8105781827 1.6596029 1.52941294 0.80577843
## 433 0.221149222 -0.7572944954 1.6596029 1.52941294 0.80577843
## 434 0.686532167 -0.7024911307 1.6596029 1.52941294 0.80577843
## 435 0.938762618 -0.7469416934 1.6596029 1.52941294 0.80577843
## 436 0.924552451 -0.7932443629 1.6596029 1.52941294 0.80577843
## 437 0.878369411 -0.8512295520 1.6596029 1.52941294 0.80577843
## 438 1.116389695 -0.8932106390 1.6596029 1.52941294 0.80577843
## 439 0.686532167 -0.9376612017 1.6596029 1.52941294 0.80577843
## 440 0.899684660 -0.9392758589 1.6596029 1.52941294 0.80577843
## 441 0.846396537 -0.9160057994 1.6596029 1.52941294 0.80577843
## 442 1.016918532 -0.8215483536 1.6596029 1.52941294 0.80577843
## 443 1.116389695 -0.8501847738 1.6596029 1.52941294 0.80577843
## 444 1.116389695 -0.8627221120 1.6596029 1.52941294 0.80577843
## 445 0.995603282 -0.9020437636 1.6596029 1.52941294 0.80577843
## 446 0.931657534 -0.8582105699 1.6596029 1.52941294 0.80577843
## 447 0.988498199 -0.8182715493 1.6596029 1.52941294 0.80577843
## 448 0.995603282 -0.7584342534 1.6596029 1.52941294 0.80577843
## 449 1.070206655 -0.7282306659 1.6596029 1.52941294 0.80577843
## 450 1.055996489 -0.7646079427 1.6596029 1.52941294 0.80577843
## 451 0.853501620 -0.6987869171 1.6596029 1.52941294 0.80577843
## 452 1.052443947 -0.6837801032 1.6596029 1.52941294 0.80577843
## 453 0.825081288 -0.6776064140 1.6596029 1.52941294 0.80577843
## 454 1.091521905 -0.6374774338 1.6596029 1.52941294 0.80577843
## 455 0.906789743 -0.6168668096 1.6596029 1.52941294 0.80577843
## 456 0.636796585 -0.6455032298 1.6596029 1.52941294 0.80577843
## 457 0.686532167 -0.5767378294 1.6596029 1.52941294 0.80577843
## 458 0.416539008 -0.4824228534 1.6596029 1.52941294 0.80577843
## 459 0.537325421 -0.4805707466 1.6596029 1.52941294 0.80577843
## 460 0.562193212 -0.5117241325 1.6596029 1.52941294 0.80577843
## 461 0.761135540 -0.5687120333 1.6596029 1.52941294 0.80577843
## 462 0.704294875 -0.5831489682 1.6596029 1.52941294 0.80577843
## 463 0.512457630 -0.5036983364 1.6596029 1.52941294 0.80577843
## 464 0.757582998 -0.4717851119 1.6596029 1.52941294 0.80577843
## 465 -0.112789684 -0.3949464255 1.6596029 1.52941294 0.80577843
## 466 -0.723826832 -0.3459843207 1.6596029 1.52941294 0.80577843
## 467 0.572850837 -0.4385896596 1.6596029 1.52941294 0.80577843
## 468 0.920999910 -0.5958762661 1.6596029 1.52941294 0.80577843
## 469 0.086152643 -0.4210658801 1.6596029 1.52941294 0.80577843
## 470 -0.421860800 -0.4612898402 1.6596029 1.52941294 0.80577843
## 471 0.547983046 -0.3617034834 1.6596029 1.52941294 0.80577843
## 472 0.786003330 -0.3304076278 1.6596029 1.52941294 0.80577843
## 473 0.228254306 -0.4267171803 1.6596029 1.52941294 0.80577843
## 474 -0.034633770 -0.5993905200 1.6596029 1.52941294 0.80577843
## 475 0.952972784 -0.6483526248 1.6596029 1.52941294 0.80577843
## 476 1.024023615 -0.7546350600 1.6596029 1.52941294 0.80577843
## 477 0.889027036 -0.7074775720 1.6596029 1.52941294 0.80577843
## 478 1.020471073 -0.8046419430 1.6596029 1.52941294 0.80577843
## 479 0.999155824 -0.7714939807 1.6596029 1.52941294 0.80577843
## 480 0.690084708 -0.8756393696 1.6596029 1.52941294 0.80577843
## 481 -0.137657475 -0.1761128861 1.6596029 1.52941294 0.80577843
## 482 0.224701764 -0.2200410597 1.6596029 1.52941294 0.80577843
## 483 0.299305137 -0.1825715149 1.6596029 1.52941294 0.80577843
## 484 -1.004477616 0.1440166471 1.6596029 1.52941294 0.80577843
## 485 -0.947636951 -0.0337381137 1.6596029 1.52941294 0.80577843
## 486 -0.592382795 0.0933923952 1.6596029 1.52941294 0.80577843
## 487 0.398776300 -0.1183176566 1.6596029 1.52941294 0.80577843
## 488 -0.546199754 -0.3052379716 1.6596029 1.52941294 0.80577843
## 489 0.857054162 -0.9375187319 -0.6373311 1.79641644 0.75958789
## 490 1.055996489 -0.9686246278 -0.6373311 1.79641644 0.75958789
## 491 1.045338864 -0.9367114033 -0.6373311 1.79641644 0.75958789
## 492 1.073759197 -0.9151034909 -0.6373311 1.79641644 0.75958789
## 493 0.530220338 -0.8002728706 -0.6373311 1.79641644 0.75958789
## 494 -0.517779422 -0.6711952751 -0.4076377 -0.10227512 0.34387304
## 495 -0.922769160 -0.6711952751 -0.4076377 -0.10227512 0.34387304
## 496 -1.413019895 -0.4732098094 -0.4076377 -0.10227512 0.34387304
## 497 0.153650933 -0.4732098094 -0.4076377 -0.10227512 0.34387304
## 498 0.071942477 -0.4285217972 -0.4076377 -0.10227512 0.34387304
## 499 -0.116342226 -0.6581830377 -0.4076377 -0.10227512 0.34387304
## 500 0.174966182 -0.6625521101 -0.4076377 -0.10227512 0.34387304
## 501 0.395223759 -0.6158695213 -0.4076377 -0.10227512 0.34387304
## 502 0.018654354 -0.6251775451 -0.9818712 -0.80241764 1.17530274
## 503 0.288647512 -0.7159307773 -0.9818712 -0.80241764 1.17530274
## 504 0.796660955 -0.7729186782 -0.9818712 -0.80241764 1.17530274
## 505 0.736267749 -0.6677760011 -0.9818712 -0.80241764 1.17530274
## 506 0.434301716 -0.6126402069 -0.9818712 -0.80241764 1.17530274
## black lstat medv
## 1 0.440615895 -1.074498970 0.159527789
## 2 0.440615895 -0.491952525 -0.101423917
## 3 0.396035074 -1.207532413 1.322937477
## 4 0.415751408 -1.360170785 1.181588636
## 5 0.440615895 -1.025486649 1.486032293
## 6 0.410165113 -1.042290874 0.670558212
## 7 0.426376321 -0.031236706 0.039924923
## 8 0.440615895 0.909799859 0.496590409
## 9 0.328123258 2.419379350 -0.655946292
## 10 0.328999540 0.622727693 -0.394994586
## 11 0.392639483 1.091845624 -0.819041108
## 12 0.440615895 0.086392864 -0.394994586
## 13 0.370513376 0.428078760 -0.090550930
## 14 0.440615895 -0.615183504 -0.231899770
## 15 0.255720500 -0.335113097 -0.471105500
## 16 0.426595391 -0.585776111 -0.286264709
## 17 0.330533033 -0.850442645 0.061670899
## 18 0.329437681 0.282442149 -0.547216415
## 19 -0.741378303 -0.134862757 -0.253645746
## 20 0.375442459 -0.192277190 -0.471105500
## 21 0.217930861 1.171665689 -0.971262937
## 22 0.392749018 0.164812578 -0.318883672
## 23 0.440615895 0.849584722 -0.797295133
## 24 0.414765591 1.012025558 -0.873406047
## 25 0.412465352 0.510699530 -0.753803182
## 26 -0.583319029 0.540106923 -0.938643973
## 27 0.221326452 0.302047077 -0.645073304
## 28 -0.550896614 0.647934029 -0.840787084
## 29 0.342472368 0.020576319 -0.449359525
## 30 0.258020739 -0.094252548 -0.166661844
## 31 0.038293155 1.392921311 -1.069119826
## 32 0.219683424 0.054184768 -0.873406047
## 33 -1.359047220 2.108501199 -1.014754888
## 34 0.022958229 0.797771697 -1.025627875
## 35 -1.186967442 1.076441751 -0.982135924
## 36 0.440615895 -0.416333515 -0.394994586
## 37 0.228774844 -0.174072614 -0.275391721
## 38 0.440615895 -0.543765550 -0.166661844
## 39 0.402607185 -0.353317674 0.235638703
## 40 0.426704926 -1.166922204 0.898890955
## 41 0.426595391 -1.494604580 1.344683452
## 42 0.314759966 -1.094103899 0.442225470
## 43 0.292414788 -0.958269752 0.300876629
## 44 0.413889309 -0.730012370 0.235638703
## 45 0.358354970 -0.434538092 -0.144915868
## 46 0.440615895 -0.342114857 -0.351502635
## 47 0.440615895 0.209623843 -0.275391721
## 48 0.395049257 0.860787538 -0.645073304
## 49 0.440615895 2.542610329 -0.884279035
## 50 0.440615895 0.496696010 -0.340629648
## 51 0.425938180 0.111599201 -0.308010684
## 52 0.408522085 -0.451342316 -0.221026782
## 53 0.440615895 -1.032488409 0.268257666
## 54 0.440615895 -0.591377519 0.094289862
## 55 0.440615895 0.300646725 -0.394994586
## 56 0.429990982 -1.098304955 1.399048391
## 57 0.440615895 -0.963871160 0.235638703
## 58 0.396801820 -1.218735230 0.985874857
## 59 0.372485009 -0.811232788 0.083416874
## 60 0.440615895 -0.480749709 -0.318883672
## 61 0.421009097 0.069588640 -0.416740562
## 62 0.234470674 0.250234052 -0.710311231
## 63 0.440615895 -0.829437365 -0.036185991
## 64 0.426157250 -0.441539852 0.268257666
## 65 0.400526017 -0.644590896 1.138096685
## 66 0.440615895 -1.117909883 0.105162850
## 67 0.440615895 -0.337913801 -0.340629648
## 68 0.433057967 -0.637589136 -0.057931966
## 69 0.440615895 0.061186528 -0.558089402
## 70 0.440615895 -0.540964846 -0.177534831
## 71 0.296358054 -0.830837717 0.181273764
## 72 0.221983663 -0.388326475 -0.090550930
## 73 0.375004318 -0.998879961 0.029051936
## 74 0.224502972 -0.716008850 0.094289862
## 75 0.418927928 -0.822435605 0.170400776
## 76 0.290881295 -0.519959566 -0.123169893
## 77 0.186056121 -0.095652900 -0.275391721
## 78 0.331737920 -0.333712745 -0.188407819
## 79 0.325603949 -0.043839875 -0.144915868
## 80 0.431414939 -0.497553933 -0.242772758
## 81 0.440615895 -1.031088057 0.594447298
## 82 0.426704926 -0.760820115 0.148654801
## 83 0.440615895 -0.830837717 0.246511690
## 84 0.372046868 -0.720209906 0.039924923
## 85 0.440615895 -0.424735627 0.148654801
## 86 0.390229709 -0.857444405 0.442225470
## 87 0.430648193 0.028978431 -0.003567028
## 88 0.421447237 -0.589977167 -0.036185991
## 89 0.440615895 -1.001680665 0.116035838
## 90 0.431414939 -0.973673624 0.670558212
## 91 0.388915287 -0.538164142 0.007305960
## 92 0.403921608 -0.623585616 -0.057931966
## 93 0.419913745 -0.629187024 0.039924923
## 94 0.434372389 -0.902255670 0.268257666
## 95 0.440615895 -0.288901480 -0.210153795
## 96 0.014304949 -0.840640181 0.637939249
## 97 0.385081555 -0.183875078 -0.123169893
## 98 0.440615895 -1.182326077 1.757856987
## 99 0.403702537 -1.271948607 2.312379361
## 100 0.440615895 -0.905056374 1.159842661
## 101 0.417175365 -0.452742668 0.540082359
## 102 0.426157250 -0.697804274 0.431352482
## 103 -3.131326538 -0.283300072 -0.427613550
## 104 0.413998845 0.110198849 -0.351502635
## 105 0.394501581 -0.045240227 -0.264518733
## 106 0.409398367 0.534505515 -0.329756660
## 107 0.427143067 0.841182610 -0.329756660
## 108 0.339733988 0.201221731 -0.231899770
## 109 0.422433054 -0.053642339 -0.297137697
## 110 0.378509444 0.405673128 -0.340629648
## 111 0.403264396 0.048583360 -0.090550930
## 112 0.426266786 -0.349116618 0.029051936
## 113 0.419256534 0.498096362 -0.405867574
## 114 0.440615895 0.621327341 -0.416740562
## 115 0.351235183 -0.308506409 -0.438486537
## 116 -0.128857540 0.435080520 -0.460232513
## 117 0.401183228 -0.085850436 -0.144915868
## 118 0.414436985 -0.329511689 -0.362375623
## 119 -0.197645637 0.380466791 -0.231899770
## 120 0.381466894 0.134004834 -0.351502635
## 121 0.355726125 0.240431588 -0.057931966
## 122 0.229979731 0.226428068 -0.242772758
## 123 0.234580209 0.738956911 -0.221026782
## 124 0.149361834 1.786420232 -0.568962390
## 125 0.248710248 0.689944590 -0.405867574
## 126 0.310488093 0.302047077 -0.123169893
## 127 0.028654058 2.045485358 -0.742930194
## 128 0.388148541 0.635330861 -0.688565255
## 129 0.440615895 0.383267495 -0.492851476
## 130 0.440615895 0.796371345 -0.895152022
## 131 0.420242350 -0.007430722 -0.362375623
## 132 0.440615895 -0.055042691 -0.318883672
## 133 0.318593697 -0.214682823 0.050797911
## 134 0.350687507 0.332854822 -0.449359525
## 135 -1.028689097 0.652135085 -0.753803182
## 136 0.416189548 0.603122764 -0.481978488
## 137 0.236332772 0.594720652 -0.558089402
## 138 0.409726972 0.271239333 -0.590708366
## 139 0.387381794 1.213676250 -1.003881900
## 140 0.440615895 0.813175569 -0.514597451
## 141 0.344005860 1.611376228 -0.927770986
## 142 0.440615895 3.046737061 -0.884279035
## 143 0.440615895 1.983869868 -0.993008912
## 144 0.440615895 1.927855787 -0.753803182
## 145 0.440615895 2.329756820 -1.166976716
## 146 -2.012862748 2.121104367 -0.949516961
## 147 -2.052733556 0.559711851 -0.753803182
## 148 0.383767133 2.363365269 -0.862533059
## 149 0.003460966 2.193922673 -0.514597451
## 150 -0.052840120 1.231880827 -0.775549157
## 151 0.176636095 0.202622083 -0.112296905
## 152 -0.165113687 0.087793216 -0.318883672
## 153 -0.146711775 -0.074647619 -0.786422145
## 154 -1.037561447 0.439281577 -0.340629648
## 155 -0.390537100 0.345457990 -0.601581353
## 156 -2.942816482 0.331454470 -0.753803182
## 157 -2.936025300 0.488293898 -1.025627875
## 158 0.074001626 -1.129112700 2.040554668
## 159 -0.030494942 -0.871447926 0.192146752
## 160 0.083640722 -0.737014131 0.083416874
## 161 -0.194469117 -1.001680665 0.485717421
## 162 0.194490331 -1.529613381 2.986504601
## 163 0.360764744 -1.503006692 2.986504601
## 164 0.348058662 -1.306957408 2.986504601
## 165 0.421009097 -0.141864517 0.018178948
## 166 -1.276238619 -0.398128939 0.268257666
## 167 0.138298780 -1.253744030 2.986504601
## 168 -1.413705278 -0.071846915 0.137781813
## 169 -0.652654802 -0.217483527 0.137781813
## 170 -0.291736362 -0.186675782 -0.025313003
## 171 -0.705231691 0.248833700 -0.558089402
## 172 -0.093587210 -0.087250788 -0.373248611
## 173 0.440615895 0.285242853 0.061670899
## 174 0.425280969 -0.505956045 0.116035838
## 175 0.400416482 -0.421934923 0.007305960
## 176 0.375551994 -1.025486649 0.746669127
## 177 0.400416482 -0.356118378 0.072543887
## 178 0.426376321 -0.891052854 0.224765715
## 179 0.378947585 -0.802830676 0.801034065
## 180 0.440615895 -1.066096858 1.594762170
## 181 0.425938180 -0.713208146 1.877459852
## 182 0.440615895 -0.448541612 1.486032293
## 183 0.410165113 -1.096904603 1.670873085
## 184 0.440615895 -0.976474328 1.083731747
## 185 0.375990135 0.185817859 0.420479494
## 186 0.333380947 0.069588640 0.768415102
## 187 0.393844370 -1.148717628 2.986504601
## 188 0.407426733 -0.836439125 1.029366808
## 189 0.286609423 -1.133313756 0.790161078
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## 505 0.402826256 -0.864446166 -0.057931966
## 506 0.440615895 -0.668396881 -1.156103728boston_scaled<-data.frame(boston_scaled, crime)Divide the dataset into train and test
n<-nrow(boston_scaled)
ind<-sample(n, size = n*0.8)
train<-boston_scaled[ind, ]
test<-boston_scaled[-ind, ]
correct_classes<-test$crimeLInear Discriminant Analysis
lda.fit<-lda(crime~., data=train)
lda.fit## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2549505 0.2301980 0.2648515 0.2500000
##
## Group means:
## zn indus chas nox rm
## low 0.99014775 -0.9159400 -0.15765625 -0.8709880 0.37229266
## med_low -0.03933548 -0.3666704 0.02401181 -0.6223415 -0.06205674
## med_high -0.39026573 0.2348380 0.24280554 0.4341211 0.06118806
## high -0.48724019 1.0171306 -0.03844192 1.0602508 -0.36424035
## age dis rad tax ptratio
## low -0.8594787 0.8995356 -0.7053471 -0.7349035 -0.40459307
## med_low -0.3681525 0.4149502 -0.5373033 -0.4786320 -0.05197491
## med_high 0.4395807 -0.4300579 -0.4205177 -0.2951387 -0.34423648
## high 0.8158306 -0.8436114 1.6379981 1.5139626 0.78062517
## black lstat medv
## low 0.37902415 -0.747428399 0.42501870
## med_low 0.31084261 -0.223672179 0.04892729
## med_high 0.08986887 0.004361028 0.16582802
## high -0.92053883 0.936378818 -0.69437854
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.07508176 0.63518169 -1.12624756
## indus 0.01502145 -0.28482992 0.18073114
## chas -0.07981630 -0.06239136 0.16457139
## nox 0.38896564 -0.75046323 -1.31343202
## rm -0.10620673 -0.04914125 -0.05026209
## age 0.24657931 -0.26154525 -0.12107561
## dis -0.08081170 -0.18204001 0.29354041
## rad 3.12450345 0.92605754 -0.19738488
## tax 0.05398304 0.02940260 0.73567524
## ptratio 0.10410133 0.03961344 -0.26805247
## black -0.13273978 0.01771556 0.08506990
## lstat 0.23965345 -0.22106990 0.43743850
## medv 0.22524316 -0.45319558 -0.05130395
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9462 0.0420 0.0118lda.arrows<-function(x, myscale=1, arrow_heads= 0.1, color="red", tex=0.75, choices =c (1,2)){
heads<-coef(x)
arrows(x0 = 0, y0=0, x1 = myscale*heads[,choices[1]],y1=myscale*heads[,choices[2]], col=color, length= arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}Convert target classes to numeric data
classes<-as.numeric(train$crime)Plot the lda results
plot(lda.fit, dimen=2, col=classes, pch=classes)
lda.arrows(lda.fit, myscale=1)
Predict the classes with the LDA model and Cross-tabulate the results
lda.pred<-predict(lda.fit, newdata = test)
table(correct=correct_classes, predict=lda.pred$class)## predict
## correct low med_low med_high high
## low 11 11 2 0
## med_low 2 20 11 0
## med_high 1 8 9 1
## high 0 0 0 26Distance measures
library(MASS)
data("Boston")
boston_scaled<-scale(Boston)
boston_scaled<-as.data.frame(boston_scaled)Euclidean distane matrix
dist_eu<-dist(boston_scaled)
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970K-means
km<-kmeans(boston_scaled, centers=3)
pairs(boston_scaled, col=km$cluster)
Investigate the optimal number of clusters
set.seed(11)
k_max<-20
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled, k)$tot.withinss})
qqplot(x=1:k_max, y = twcss, geom="Line")## Warning in plot.window(...): "geom" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "geom" is not a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "geom" is not
## a graphical parameter
## Warning in axis(side = side, at = at, labels = labels, ...): "geom" is not
## a graphical parameter## Warning in box(...): "geom" is not a graphical parameter## Warning in title(...): "geom" is not a graphical parameter
Perform k-means with different number of clusters
km<-kmeans(boston_scaled, centers=6)
pairs(boston_scaled, col = km$cluster)
library(GGally)## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2##
## Attaching package: 'GGally'## The following object is masked from 'package:dplyr':
##
## nasaggpairs(boston_scaled)
Super Bonus
model_predictors<- dplyr::select(train, -crime)Check th dimensions
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3Matrix multiplication
matrix_product <- as.matrix(model_predictors)%*%lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)Create a 3D plot
library(plotly)##
## Attaching package: 'plotly'## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layoutplot_ly(x=matrix_product$LD1, y=matrix_product$LD2, z=matrix_product$LD3, type="scatter3d", mode="markers")plot_ly(x=matrix_product$LD1, y=matrix_product$LD2, z=matrix_product$LD3, type="scatter3d", mode="markers", color = train$crime)km <-kmeans(boston_scaled, centers = 6)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = km$cluster[ind])Dimensionality Reduction Techniques (Excercise 5)
Human data
###1.Read and Take a look at data
human <- read.table("http://s3.amazonaws.com/assets.datacamp.com/production/course_2218/datasets/human2.txt", sep=",", header=T)
library(MASS);library(tidyr);library(dplyr);library(GGally);library(ggplot2);library(dplyr);library(tidyr);library(corrplot)
str(human);dim(human);summary(human)## 'data.frame': 155 obs. of 8 variables:
## $ Edu2.FM : num 1.007 0.997 0.983 0.989 0.969 ...
## $ Labo.FM : num 0.891 0.819 0.825 0.884 0.829 ...
## $ Edu.Exp : num 17.5 20.2 15.8 18.7 17.9 16.5 18.6 16.5 15.9 19.2 ...
## $ Life.Exp : num 81.6 82.4 83 80.2 81.6 80.9 80.9 79.1 82 81.8 ...
## $ GNI : int 64992 42261 56431 44025 45435 43919 39568 52947 42155 32689 ...
## $ Mat.Mor : int 4 6 6 5 6 7 9 28 11 8 ...
## $ Ado.Birth: num 7.8 12.1 1.9 5.1 6.2 3.8 8.2 31 14.5 25.3 ...
## $ Parli.F : num 39.6 30.5 28.5 38 36.9 36.9 19.9 19.4 28.2 31.4 ...## [1] 155 8## Edu2.FM Labo.FM Edu.Exp Life.Exp
## Min. :0.1717 Min. :0.1857 Min. : 5.40 Min. :49.00
## 1st Qu.:0.7264 1st Qu.:0.5984 1st Qu.:11.25 1st Qu.:66.30
## Median :0.9375 Median :0.7535 Median :13.50 Median :74.20
## Mean :0.8529 Mean :0.7074 Mean :13.18 Mean :71.65
## 3rd Qu.:0.9968 3rd Qu.:0.8535 3rd Qu.:15.20 3rd Qu.:77.25
## Max. :1.4967 Max. :1.0380 Max. :20.20 Max. :83.50
## GNI Mat.Mor Ado.Birth Parli.F
## Min. : 581 Min. : 1.0 Min. : 0.60 Min. : 0.00
## 1st Qu.: 4198 1st Qu.: 11.5 1st Qu.: 12.65 1st Qu.:12.40
## Median : 12040 Median : 49.0 Median : 33.60 Median :19.30
## Mean : 17628 Mean : 149.1 Mean : 47.16 Mean :20.91
## 3rd Qu.: 24512 3rd Qu.: 190.0 3rd Qu.: 71.95 3rd Qu.:27.95
## Max. :123124 Max. :1100.0 Max. :204.80 Max. :57.50155 oberservations and 8 variables
To see structure and relationships
cor(human)## Edu2.FM Labo.FM Edu.Exp Life.Exp GNI
## Edu2.FM 1.000000000 0.009564039 0.59325156 0.5760299 0.43030485
## Labo.FM 0.009564039 1.000000000 0.04732183 -0.1400125 -0.02173971
## Edu.Exp 0.593251562 0.047321827 1.00000000 0.7894392 0.62433940
## Life.Exp 0.576029853 -0.140012504 0.78943917 1.0000000 0.62666411
## GNI 0.430304846 -0.021739705 0.62433940 0.6266641 1.00000000
## Mat.Mor -0.660931770 0.240461075 -0.73570257 -0.8571684 -0.49516234
## Ado.Birth -0.529418415 0.120158862 -0.70356489 -0.7291774 -0.55656208
## Parli.F 0.078635285 0.250232608 0.20608156 0.1700863 0.08920818
## Mat.Mor Ado.Birth Parli.F
## Edu2.FM -0.6609318 -0.5294184 0.07863528
## Labo.FM 0.2404611 0.1201589 0.25023261
## Edu.Exp -0.7357026 -0.7035649 0.20608156
## Life.Exp -0.8571684 -0.7291774 0.17008631
## GNI -0.4951623 -0.5565621 0.08920818
## Mat.Mor 1.0000000 0.7586615 -0.08944000
## Ado.Birth 0.7586615 1.0000000 -0.07087810
## Parli.F -0.0894400 -0.0708781 1.00000000ggpairs(human)
cor(human)%>%corrplot(method="circle", type="upper", title="Graph 1. Correlations between variables in the human data")
- Hypothesis
- negative relationship between Edu2.FM and Mat.Mor
- negative relationship between Edu.Exp and Mat.Mor
- negative relationship between Life.Exp and Mat.Mor
- negative relationship between GNI and Mat.Mor
2.Principal component analysis
pca_human <- prcomp(human)
pca_human## Standard deviations (1, .., p=8):
## [1] 1.854416e+04 1.855219e+02 2.518701e+01 1.145441e+01 3.766241e+00
## [6] 1.565912e+00 1.912052e-01 1.591112e-01
##
## Rotation (n x k) = (8 x 8):
## PC1 PC2 PC3 PC4
## Edu2.FM -5.607472e-06 0.0006713951 -3.412027e-05 -2.736326e-04
## Labo.FM 2.331945e-07 -0.0002819357 5.302884e-04 -4.692578e-03
## Edu.Exp -9.562910e-05 0.0075529759 1.427664e-02 -3.313505e-02
## Life.Exp -2.815823e-04 0.0283150248 1.294971e-02 -6.752684e-02
## GNI -9.999832e-01 -0.0057723054 -5.156742e-04 4.932889e-05
## Mat.Mor 5.655734e-03 -0.9916320120 1.260302e-01 -6.100534e-03
## Ado.Birth 1.233961e-03 -0.1255502723 -9.918113e-01 5.301595e-03
## Parli.F -5.526460e-05 0.0032317269 -7.398331e-03 -9.971232e-01
## PC5 PC6 PC7 PC8
## Edu2.FM -0.0022935252 2.180183e-02 6.998623e-01 7.139410e-01
## Labo.FM 0.0022190154 3.264423e-02 7.132267e-01 -7.001533e-01
## Edu.Exp 0.1431180282 9.882477e-01 -3.826887e-02 7.776451e-03
## Life.Exp 0.9865644425 -1.453515e-01 5.380452e-03 2.281723e-03
## GNI -0.0001135863 -2.711698e-05 -8.075191e-07 -1.176762e-06
## Mat.Mor 0.0266373214 1.695203e-03 1.355518e-04 8.371934e-04
## Ado.Birth 0.0188618600 1.273198e-02 -8.641234e-05 -1.707885e-04
## Parli.F -0.0716401914 -2.309896e-02 -2.642548e-03 2.680113e-03biplot(pca_human, choices = 1:2, cex = c(0.8, 1), col= c("grey40", "deeppink2"), main = "Grpagh 2. PCA with human data, unscaled")## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
human.std <- scale(human)
pca_human2 <- prcomp(human.std)
biplot(pca_human2, choices= 1:2, cex = c(0.8, 1), col= c("grey40", "deeppink2"), main="Graph 3. PCA with human data, scaled")
s <- summary(pca_human2)
pca_pr <- round(100*s$importance[2, ], digits = 1)pc_lab <- paste0(names(pca_pr), " (", pca_pr, "%)")biplot(pca_human2, cex = c(0.8, 1), col = c("grey40", "deeppink2"), xlab = pc_lab[1], ylab = pc_lab[2], main="Graph 4. PCA with human data, scaled, %")
library(FactoMineR)
es.pca = PCA(human, scale.unit= TRUE, ncp=5)
tea dataset
library(ggplot2)
library(dplyr)
library(tidyr)
library(FactoMineR)data(tea)
my_tea <- data.frame(tea)
dim(my_tea)## [1] 300 36str(my_tea)## 'data.frame': 300 obs. of 36 variables:
## $ breakfast : Factor w/ 2 levels "breakfast","Not.breakfast": 1 1 2 2 1 2 1 2 1 1 ...
## $ tea.time : Factor w/ 2 levels "Not.tea time",..: 1 1 2 1 1 1 2 2 2 1 ...
## $ evening : Factor w/ 2 levels "evening","Not.evening": 2 2 1 2 1 2 2 1 2 1 ...
## $ lunch : Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
## $ dinner : Factor w/ 2 levels "dinner","Not.dinner": 2 2 1 1 2 1 2 2 2 2 ...
## $ always : Factor w/ 2 levels "always","Not.always": 2 2 2 2 1 2 2 2 2 2 ...
## $ home : Factor w/ 2 levels "home","Not.home": 1 1 1 1 1 1 1 1 1 1 ...
## $ work : Factor w/ 2 levels "Not.work","work": 1 1 2 1 1 1 1 1 1 1 ...
## $ tearoom : Factor w/ 2 levels "Not.tearoom",..: 1 1 1 1 1 1 1 1 1 2 ...
## $ friends : Factor w/ 2 levels "friends","Not.friends": 2 2 1 2 2 2 1 2 2 2 ...
## $ resto : Factor w/ 2 levels "Not.resto","resto": 1 1 2 1 1 1 1 1 1 1 ...
## $ pub : Factor w/ 2 levels "Not.pub","pub": 1 1 1 1 1 1 1 1 1 1 ...
## $ Tea : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
## $ How : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
## $ sugar : Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
## $ how : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ where : Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ price : Factor w/ 6 levels "p_branded","p_cheap",..: 4 6 6 6 6 3 6 6 5 5 ...
## $ age : int 39 45 47 23 48 21 37 36 40 37 ...
## $ sex : Factor w/ 2 levels "F","M": 2 1 1 2 2 2 2 1 2 2 ...
## $ SPC : Factor w/ 7 levels "employee","middle",..: 2 2 4 6 1 6 5 2 5 5 ...
## $ Sport : Factor w/ 2 levels "Not.sportsman",..: 2 2 2 1 2 2 2 2 2 1 ...
## $ age_Q : Factor w/ 5 levels "15-24","25-34",..: 3 4 4 1 4 1 3 3 3 3 ...
## $ frequency : Factor w/ 4 levels "1/day","1 to 2/week",..: 1 1 3 1 3 1 4 2 3 3 ...
## $ escape.exoticism: Factor w/ 2 levels "escape-exoticism",..: 2 1 2 1 1 2 2 2 2 2 ...
## $ spirituality : Factor w/ 2 levels "Not.spirituality",..: 1 1 1 2 2 1 1 1 1 1 ...
## $ healthy : Factor w/ 2 levels "healthy","Not.healthy": 1 1 1 1 2 1 1 1 2 1 ...
## $ diuretic : Factor w/ 2 levels "diuretic","Not.diuretic": 2 1 1 2 1 2 2 2 2 1 ...
## $ friendliness : Factor w/ 2 levels "friendliness",..: 2 2 1 2 1 2 2 1 2 1 ...
## $ iron.absorption : Factor w/ 2 levels "iron absorption",..: 2 2 2 2 2 2 2 2 2 2 ...
## $ feminine : Factor w/ 2 levels "feminine","Not.feminine": 2 2 2 2 2 2 2 1 2 2 ...
## $ sophisticated : Factor w/ 2 levels "Not.sophisticated",..: 1 1 1 2 1 1 1 2 2 1 ...
## $ slimming : Factor w/ 2 levels "No.slimming",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ exciting : Factor w/ 2 levels "exciting","No.exciting": 2 1 2 2 2 2 2 2 2 2 ...
## $ relaxing : Factor w/ 2 levels "No.relaxing",..: 1 1 2 2 2 2 2 2 2 2 ...
## $ effect.on.health: Factor w/ 2 levels "effect on health",..: 2 2 2 2 2 2 2 2 2 2 ...summary(my_tea)## breakfast tea.time evening lunch
## breakfast :144 Not.tea time:131 evening :103 lunch : 44
## Not.breakfast:156 tea time :169 Not.evening:197 Not.lunch:256
##
##
##
##
##
## dinner always home work
## dinner : 21 always :103 home :291 Not.work:213
## Not.dinner:279 Not.always:197 Not.home: 9 work : 87
##
##
##
##
##
## tearoom friends resto pub
## Not.tearoom:242 friends :196 Not.resto:221 Not.pub:237
## tearoom : 58 Not.friends:104 resto : 79 pub : 63
##
##
##
##
##
## Tea How sugar how
## black : 74 alone:195 No.sugar:155 tea bag :170
## Earl Grey:193 lemon: 33 sugar :145 tea bag+unpackaged: 94
## green : 33 milk : 63 unpackaged : 36
## other: 9
##
##
##
## where price age sex
## chain store :192 p_branded : 95 Min. :15.00 F:178
## chain store+tea shop: 78 p_cheap : 7 1st Qu.:23.00 M:122
## tea shop : 30 p_private label: 21 Median :32.00
## p_unknown : 12 Mean :37.05
## p_upscale : 53 3rd Qu.:48.00
## p_variable :112 Max. :90.00
##
## SPC Sport age_Q frequency
## employee :59 Not.sportsman:121 15-24:92 1/day : 95
## middle :40 sportsman :179 25-34:69 1 to 2/week: 44
## non-worker :64 35-44:40 +2/day :127
## other worker:20 45-59:61 3 to 6/week: 34
## senior :35 +60 :38
## student :70
## workman :12
## escape.exoticism spirituality healthy
## escape-exoticism :142 Not.spirituality:206 healthy :210
## Not.escape-exoticism:158 spirituality : 94 Not.healthy: 90
##
##
##
##
##
## diuretic friendliness iron.absorption
## diuretic :174 friendliness :242 iron absorption : 31
## Not.diuretic:126 Not.friendliness: 58 Not.iron absorption:269
##
##
##
##
##
## feminine sophisticated slimming
## feminine :129 Not.sophisticated: 85 No.slimming:255
## Not.feminine:171 sophisticated :215 slimming : 45
##
##
##
##
##
## exciting relaxing effect.on.health
## exciting :116 No.relaxing:113 effect on health : 66
## No.exciting:184 relaxing :187 No.effect on health:234
##
##
##
##
## gather(my_tea) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free") + geom_bar() + theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 8))## Warning: attributes are not identical across measure variables;
## they will be dropped
keep_columns <- c("breakfast", "sex", "price", "healthy", "spirituality", "friends", "Tea", "tearoom")
my_tea1 <- dplyr::select(my_tea, one_of(keep_columns))
gather(my_tea1) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free") + geom_bar() + theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 8))## Warning: attributes are not identical across measure variables;
## they will be dropped
3. Multiple Correspondence Analysis
mca <- MCA(my_tea1, graph = FALSE)
summary(mca)##
## Call:
## MCA(X = my_tea1, graph = FALSE)
##
##
## Eigenvalues
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6
## Variance 0.193 0.165 0.150 0.147 0.139 0.128
## % of var. 11.885 10.164 9.218 9.031 8.544 7.859
## Cumulative % of var. 11.885 22.050 31.268 40.299 48.843 56.702
## Dim.7 Dim.8 Dim.9 Dim.10 Dim.11 Dim.12
## Variance 0.121 0.115 0.110 0.102 0.096 0.087
## % of var. 7.429 7.054 6.784 6.297 5.928 5.371
## Cumulative % of var. 64.131 71.185 77.970 84.267 90.195 95.565
## Dim.13
## Variance 0.072
## % of var. 4.435
## Cumulative % of var. 100.000
##
## Individuals (the 10 first)
## Dim.1 ctr cos2 Dim.2 ctr cos2 Dim.3
## 1 | 0.767 1.015 0.144 | 0.347 0.242 0.029 | 0.713
## 2 | 0.261 0.117 0.057 | 0.285 0.164 0.069 | -0.327
## 3 | -0.400 0.276 0.233 | -0.236 0.113 0.081 | 0.005
## 4 | 0.057 0.006 0.003 | -0.199 0.080 0.034 | 0.209
## 5 | -0.005 0.000 0.000 | 0.067 0.009 0.003 | 0.484
## 6 | 0.641 0.709 0.171 | -0.080 0.013 0.003 | 0.234
## 7 | -0.151 0.039 0.028 | 0.019 0.001 0.000 | 0.020
## 8 | 0.262 0.118 0.059 | 0.161 0.052 0.022 | -0.149
## 9 | 0.533 0.491 0.180 | 0.465 0.437 0.137 | 0.375
## 10 | 0.503 0.436 0.118 | 1.130 2.577 0.594 | -0.477
## ctr cos2
## 1 1.132 0.125 |
## 2 0.238 0.090 |
## 3 0.000 0.000 |
## 4 0.097 0.037 |
## 5 0.521 0.164 |
## 6 0.122 0.023 |
## 7 0.001 0.001 |
## 8 0.050 0.019 |
## 9 0.313 0.089 |
## 10 0.505 0.106 |
##
## Categories (the 10 first)
## Dim.1 ctr cos2 v.test Dim.2 ctr
## breakfast | -0.002 0.000 0.000 -0.030 | 0.211 1.613
## Not.breakfast | 0.002 0.000 0.000 0.030 | -0.194 1.489
## F | -0.357 4.897 0.186 -7.458 | -0.173 1.347
## M | 0.521 7.144 0.186 7.458 | 0.253 1.965
## p_branded | 0.289 1.708 0.039 3.398 | -0.960 22.087
## p_cheap | 0.678 0.694 0.011 1.812 | 0.329 0.191
## p_private label | 0.723 2.369 0.039 3.431 | 0.274 0.398
## p_unknown | 0.192 0.095 0.002 0.677 | -0.040 0.005
## p_upscale | 0.563 3.621 0.068 4.507 | 1.183 18.709
## p_variable | -0.710 12.168 0.300 -9.471 | 0.187 0.986
## cos2 v.test Dim.3 ctr cos2 v.test
## breakfast 0.041 3.500 | -0.286 3.276 0.075 -4.751 |
## Not.breakfast 0.041 -3.500 | 0.264 3.024 0.075 4.751 |
## F 0.044 -3.618 | -0.242 2.905 0.086 -5.059 |
## M 0.044 3.618 | 0.353 4.238 0.086 5.059 |
## p_branded 0.427 -11.300 | -0.341 3.069 0.054 -4.012 |
## p_cheap 0.003 0.880 | 1.264 3.110 0.038 3.378 |
## p_private label 0.006 1.301 | 0.150 0.131 0.002 0.709 |
## p_unknown 0.000 -0.142 | 2.649 23.430 0.292 9.351 |
## p_upscale 0.300 9.475 | -0.267 1.049 0.015 -2.137 |
## p_variable 0.021 2.493 | 0.024 0.019 0.000 0.326 |
##
## Categorical variables (eta2)
## Dim.1 Dim.2 Dim.3
## breakfast | 0.000 0.041 0.075 |
## sex | 0.186 0.044 0.086 |
## price | 0.319 0.560 0.369 |
## healthy | 0.010 0.044 0.413 |
## spirituality | 0.082 0.019 0.000 |
## friends | 0.412 0.000 0.000 |
## Tea | 0.272 0.340 0.162 |
## tearoom | 0.264 0.273 0.092 |plot(mca, invisible=c("ind"), habillage="quali")
res.mca=MCA(my_tea1)
Analysis of longitudinal data (Exercise 6)
Read data
BPRS<-read.table("https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/BPRS.txt", sep=" ", header=TRUE)
RATS<- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/rats.txt", sep="\t", header=TRUE)
library(ggplot2);library(dplyr);library(tidyr)Task 1
Convert catergorial variables in RATS dataset to factors and to long form
RATS$Group<-factor(RATS$Group)
RATS$ID<-factor(RATS$ID)
glimpse(RATS)## Observations: 16
## Variables: 13
## $ ID <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3
## $ WD1 <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, 5...
## $ WD8 <int> 250, 230, 250, 255, 260, 265, 275, 255, 415, 420, 445, 5...
## $ WD15 <int> 255, 230, 250, 255, 255, 270, 260, 260, 425, 430, 450, 5...
## $ WD22 <int> 260, 232, 255, 265, 270, 275, 270, 268, 428, 440, 452, 5...
## $ WD29 <int> 262, 240, 262, 265, 270, 275, 273, 270, 438, 448, 455, 5...
## $ WD36 <int> 258, 240, 265, 268, 273, 277, 274, 265, 443, 460, 455, 5...
## $ WD43 <int> 266, 243, 267, 270, 274, 278, 276, 265, 442, 458, 451, 5...
## $ WD44 <int> 266, 244, 267, 272, 273, 278, 271, 267, 446, 464, 450, 5...
## $ WD50 <int> 265, 238, 264, 274, 276, 284, 282, 273, 456, 475, 462, 6...
## $ WD57 <int> 272, 247, 268, 273, 278, 279, 281, 274, 468, 484, 466, 6...
## $ WD64 <int> 278, 245, 269, 275, 280, 281, 284, 278, 478, 496, 472, 6...head(RATS)## ID Group WD1 WD8 WD15 WD22 WD29 WD36 WD43 WD44 WD50 WD57 WD64
## 1 1 1 240 250 255 260 262 258 266 266 265 272 278
## 2 2 1 225 230 230 232 240 240 243 244 238 247 245
## 3 3 1 245 250 250 255 262 265 267 267 264 268 269
## 4 4 1 260 255 255 265 265 268 270 272 274 273 275
## 5 5 1 255 260 255 270 270 273 274 273 276 278 280
## 6 6 1 260 265 270 275 275 277 278 278 284 279 281RATSL<-gather(RATS, key=WDs, value=weight, WD1:WD64)%>%mutate(time= as.integer(substr(WDs,3,4)))
glimpse(RATSL)## Observations: 176
## Variables: 5
## $ ID <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 1...
## $ WDs <chr> "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1",...
## $ weight <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, ...
## $ time <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 8, 8...head(RATSL)## ID Group WDs weight time
## 1 1 1 WD1 240 1
## 2 2 1 WD1 225 1
## 3 3 1 WD1 245 1
## 4 4 1 WD1 260 1
## 5 5 1 WD1 255 1
## 6 6 1 WD1 260 1tail(RATSL)## ID Group WDs weight time
## 171 11 2 WD64 472 64
## 172 12 2 WD64 628 64
## 173 13 3 WD64 525 64
## 174 14 3 WD64 559 64
## 175 15 3 WD64 548 64
## 176 16 3 WD64 569 64ggplot(RATSL, aes(x = time, y = weight, linetype = ID)) +
geom_line() +
scale_linetype_manual(values = rep(1:10, times=4)) +
facet_grid(. ~ Group, labeller = label_both) +
theme(legend.position = "none") +
scale_y_continuous(limits = c(min(RATSL$weight), max(RATSL$weight)))
Standardise the weight
RATSL<- RATSL%>%group_by(time)%>%mutate(stdweight = (weight - mean(weight))/sd(weight))%>%ungroup()
glimpse(RATSL)## Observations: 176
## Variables: 6
## $ ID <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1...
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1...
## $ WDs <chr> "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD...
## $ weight <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 44...
## $ time <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 8...
## $ stdweight <dbl> -1.0011429, -1.1203857, -0.9613953, -0.8421525, -0.8...p1<-ggplot(RATSL, aes(x=time, y=stdweight, linetype=ID))
p2<-p1+geom_line()+scale_linetype_manual(values=rep(1:10, times=4))
p3<-p2+facet_grid(. ~Group, labeller = label_both)
p4<-p3+theme_bw()+theme(legend.position ="none")
p5<-p4+theme(panel.grid.minor.y=element_blank())
p6<-p5+scale_y_continuous(name="standardized weights")
p6
Numberof times, baseline (time 0)
n<-RATSL$time%>%unique()%>%length()Make a summary data
RATSS<-RATSL%>%group_by(Group, time)%>% summarise(mean=mean(weight), se=sd(weight)/sqrt(n))%>%ungroup()
glimpse(RATSS)## Observations: 33
## Variables: 4
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2,...
## $ time <int> 1, 8, 15, 22, 29, 36, 43, 44, 50, 57, 64, 1, 8, 15, 22, ...
## $ mean <dbl> 250.625, 255.000, 254.375, 261.875, 264.625, 265.000, 26...
## $ se <dbl> 4.589478, 3.947710, 3.460116, 4.100800, 3.333956, 3.5529...Visualize the mean weight and SE
p1<-ggplot(RATSS, aes(x=time, y=mean, linetype=Group, shape=Group))
p2<-p1+geom_line()+scale_linetype_manual(values=c(1,2,3))
p3<-p2+geom_point(size=3)+scale_shape_manual(values=c(1,2,3))
p4<-p3+geom_errorbar(aes(ymin=mean-se, ymax=mean+se, linetype="1"), width=0.3)
p5<-p4+theme_bw()+theme(panel.grid.major=element_blank(), panel.grid.minor=element_blank())
p6<-p5+theme(legend.position= "top")
p7<-p6+scale_y_continuous(name="mean(weights) +/- se(weights)")
p7
Task 2
BPRS$treatment<-factor(BPRS$treatment)
BPRS$subject<-factor(BPRS$subject)
str(BPRS)## 'data.frame': 40 obs. of 11 variables:
## $ treatment: Factor w/ 2 levels "1","2": 1 1 1 1 1 1 1 1 1 1 ...
## $ subject : Factor w/ 20 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ...
## $ week0 : int 42 58 54 55 72 48 71 30 41 57 ...
## $ week1 : int 36 68 55 77 75 43 61 36 43 51 ...
## $ week2 : int 36 61 41 49 72 41 47 38 39 51 ...
## $ week3 : int 43 55 38 54 65 38 30 38 35 55 ...
## $ week4 : int 41 43 43 56 50 36 27 31 28 53 ...
## $ week5 : int 40 34 28 50 39 29 40 26 22 43 ...
## $ week6 : int 38 28 29 47 32 33 30 26 20 43 ...
## $ week7 : int 47 28 25 42 38 27 31 25 23 39 ...
## $ week8 : int 51 28 24 46 32 25 31 24 21 32 ...BPRSL<-gather(BPRS, key=weeks, value=bprs, -treatment, -subject)%>%mutate(week=as.integer(substr(weeks, 5, 5)))
glimpse(BPRSL)## Observations: 360
## Variables: 5
## $ treatment <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
## $ subject <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1...
## $ weeks <chr> "week0", "week0", "week0", "week0", "week0", "week0"...
## $ bprs <int> 42, 58, 54, 55, 72, 48, 71, 30, 41, 57, 30, 55, 36, ...
## $ week <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...head(BPRSL)## treatment subject weeks bprs week
## 1 1 1 week0 42 0
## 2 1 2 week0 58 0
## 3 1 3 week0 54 0
## 4 1 4 week0 55 0
## 5 1 5 week0 72 0
## 6 1 6 week0 48 0tail(BPRSL)## treatment subject weeks bprs week
## 355 2 15 week8 37 8
## 356 2 16 week8 22 8
## 357 2 17 week8 43 8
## 358 2 18 week8 30 8
## 359 2 19 week8 21 8
## 360 2 20 week8 23 8BPRS_reg<-lm(bprs ~ week + treatment, data=BPRSL)
summary(BPRS_reg)##
## Call:
## lm(formula = bprs ~ week + treatment, data = BPRSL)
##
## Residuals:
## Min 1Q Median 3Q Max
## -22.454 -8.965 -3.196 7.002 50.244
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 46.4539 1.3670 33.982 <2e-16 ***
## week -2.2704 0.2524 -8.995 <2e-16 ***
## treatment2 0.5722 1.3034 0.439 0.661
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12.37 on 357 degrees of freedom
## Multiple R-squared: 0.1851, Adjusted R-squared: 0.1806
## F-statistic: 40.55 on 2 and 357 DF, p-value: < 2.2e-16library(lme4)## Loading required package: Matrix##
## Attaching package: 'Matrix'## The following objects are masked from 'package:tidyr':
##
## expand, pack, unpackBPRS_ref<-lmer(bprs ~ week + treatment + (1|subject), data=BPRSL, REML=F)
summary(BPRS_ref)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: bprs ~ week + treatment + (1 | subject)
## Data: BPRSL
##
## AIC BIC logLik deviance df.resid
## 2748.7 2768.1 -1369.4 2738.7 355
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.0481 -0.6749 -0.1361 0.4813 3.4855
##
## Random effects:
## Groups Name Variance Std.Dev.
## subject (Intercept) 47.41 6.885
## Residual 104.21 10.208
## Number of obs: 360, groups: subject, 20
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 46.4539 1.9090 24.334
## week -2.2704 0.2084 -10.896
## treatment2 0.5722 1.0761 0.532
##
## Correlation of Fixed Effects:
## (Intr) week
## week -0.437
## treatment2 -0.282 0.000BPRS_ref1<-lmer(bprs~week+treatment+(week|subject), data=BPRSL, REML=F)
summary(BPRS_ref1)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: bprs ~ week + treatment + (week | subject)
## Data: BPRSL
##
## AIC BIC logLik deviance df.resid
## 2745.4 2772.6 -1365.7 2731.4 353
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.8919 -0.6194 -0.0691 0.5531 3.7977
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## subject (Intercept) 64.8222 8.0512
## week 0.9609 0.9803 -0.51
## Residual 97.4304 9.8707
## Number of obs: 360, groups: subject, 20
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 46.4539 2.1052 22.066
## week -2.2704 0.2977 -7.626
## treatment2 0.5722 1.0405 0.550
##
## Correlation of Fixed Effects:
## (Intr) week
## week -0.582
## treatment2 -0.247 0.000anova(BPRS_ref1, BPRS_ref)## Data: BPRSL
## Models:
## BPRS_ref: bprs ~ week + treatment + (1 | subject)
## BPRS_ref1: bprs ~ week + treatment + (week | subject)
## Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
## BPRS_ref 5 2748.7 2768.1 -1369.4 2738.7
## BPRS_ref1 7 2745.4 2772.6 -1365.7 2731.4 7.2721 2 0.02636 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1BPRS_ref2<-lmer(bprs~week+treatment+(treatment|subject), data=BPRSL, REML=F)
summary(BPRS_ref2)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: bprs ~ week + treatment + (treatment | subject)
## Data: BPRSL
##
## AIC BIC logLik deviance df.resid
## 2584.7 2611.9 -1285.4 2570.7 353
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.3048 -0.6022 -0.0652 0.4414 3.1690
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## subject (Intercept) 64.43 8.027
## treatment2 188.83 13.742 -0.56
## Residual 54.23 7.364
## Number of obs: 360, groups: subject, 20
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 46.4539 1.9708 23.571
## week -2.2704 0.1503 -15.104
## treatment2 0.5722 3.1692 0.181
##
## Correlation of Fixed Effects:
## (Intr) week
## week -0.305
## treatment2 -0.540 0.000anova(BPRS_ref1, BPRS_ref2)## Data: BPRSL
## Models:
## BPRS_ref1: bprs ~ week + treatment + (week | subject)
## BPRS_ref2: bprs ~ week + treatment + (treatment | subject)
## Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
## BPRS_ref1 7 2745.4 2772.6 -1365.7 2731.4
## BPRS_ref2 7 2584.7 2611.9 -1285.4 2570.7 160.7 0 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Fitted<-fitted(BPRS_ref2)
BPRSL%>% mutate(Fitted)## treatment subject weeks bprs week Fitted
## 1 1 1 week0 42 0 50.29960
## 2 1 2 week0 58 0 53.19735
## 3 1 3 week0 54 0 46.50943
## 4 1 4 week0 55 0 60.62454
## 5 1 5 week0 72 0 60.56119
## 6 1 6 week0 48 0 44.77537
## 7 1 7 week0 71 0 49.67670
## 8 1 8 week0 30 0 40.15001
## 9 1 9 week0 41 0 39.89941
## 10 1 10 week0 57 0 55.36484
## 11 1 11 week0 30 0 44.95909
## 12 1 12 week0 55 0 53.41261
## 13 1 13 week0 36 0 37.39683
## 14 1 14 week0 38 0 39.90752
## 15 1 15 week0 66 0 53.93870
## 16 1 16 week0 41 0 43.76270
## 17 1 17 week0 45 0 45.30333
## 18 1 18 week0 39 0 37.56531
## 19 1 19 week0 24 0 35.43005
## 20 1 20 week0 38 0 36.34320
## 21 2 1 week0 52 0 57.49443
## 22 2 2 week0 30 0 32.65015
## 23 2 3 week0 65 0 41.66325
## 24 2 4 week0 37 0 37.97522
## 25 2 5 week0 59 0 57.51543
## 26 2 6 week0 30 0 38.15519
## 27 2 7 week0 69 0 50.69652
## 28 2 8 week0 62 0 62.78354
## 29 2 9 week0 38 0 38.57000
## 30 2 10 week0 65 0 49.96478
## 31 2 11 week0 78 0 80.10357
## 32 2 12 week0 38 0 38.81005
## 33 2 13 week0 63 0 57.68042
## 34 2 14 week0 40 0 42.71172
## 35 2 15 week0 40 0 48.05030
## 36 2 16 week0 54 0 39.85227
## 37 2 17 week0 33 0 48.35123
## 38 2 18 week0 28 0 39.94579
## 39 2 19 week0 52 0 39.09184
## 40 2 20 week0 47 0 38.45652
## 41 1 1 week1 36 1 48.02919
## 42 1 2 week1 68 1 50.92693
## 43 1 3 week1 55 1 44.23901
## 44 1 4 week1 77 1 58.35412
## 45 1 5 week1 75 1 58.29078
## 46 1 6 week1 43 1 42.50495
## 47 1 7 week1 61 1 47.40628
## 48 1 8 week1 36 1 37.87959
## 49 1 9 week1 43 1 37.62899
## 50 1 10 week1 51 1 53.09442
## 51 1 11 week1 34 1 42.68868
## 52 1 12 week1 52 1 51.14219
## 53 1 13 week1 32 1 35.12642
## 54 1 14 week1 35 1 37.63710
## 55 1 15 week1 68 1 51.66828
## 56 1 16 week1 35 1 41.49228
## 57 1 17 week1 38 1 43.03291
## 58 1 18 week1 35 1 35.29490
## 59 1 19 week1 28 1 33.15963
## 60 1 20 week1 34 1 34.07279
## 61 2 1 week1 73 1 55.22402
## 62 2 2 week1 23 1 30.37974
## 63 2 3 week1 31 1 39.39284
## 64 2 4 week1 31 1 35.70481
## 65 2 5 week1 67 1 55.24501
## 66 2 6 week1 33 1 35.88477
## 67 2 7 week1 52 1 48.42610
## 68 2 8 week1 54 1 60.51312
## 69 2 9 week1 40 1 36.29958
## 70 2 10 week1 44 1 47.69436
## 71 2 11 week1 95 1 77.83315
## 72 2 12 week1 41 1 36.53963
## 73 2 13 week1 65 1 55.41001
## 74 2 14 week1 37 1 40.44130
## 75 2 15 week1 36 1 45.77989
## 76 2 16 week1 45 1 37.58186
## 77 2 17 week1 41 1 46.08081
## 78 2 18 week1 30 1 37.67537
## 79 2 19 week1 43 1 36.82142
## 80 2 20 week1 36 1 36.18611
## 81 1 1 week2 36 2 45.75877
## 82 1 2 week2 61 2 48.65652
## 83 1 3 week2 41 2 41.96860
## 84 1 4 week2 49 2 56.08371
## 85 1 5 week2 72 2 56.02036
## 86 1 6 week2 41 2 40.23454
## 87 1 7 week2 47 2 45.13587
## 88 1 8 week2 38 2 35.60918
## 89 1 9 week2 39 2 35.35857
## 90 1 10 week2 51 2 50.82400
## 91 1 11 week2 34 2 40.41826
## 92 1 12 week2 49 2 48.87178
## 93 1 13 week2 36 2 32.85600
## 94 1 14 week2 36 2 35.36668
## 95 1 15 week2 65 2 49.39786
## 96 1 16 week2 45 2 39.22186
## 97 1 17 week2 46 2 40.76250
## 98 1 18 week2 27 2 33.02448
## 99 1 19 week2 31 2 30.88922
## 100 1 20 week2 27 2 31.80237
## 101 2 1 week2 42 2 52.95360
## 102 2 2 week2 32 2 28.10932
## 103 2 3 week2 33 2 37.12242
## 104 2 4 week2 27 2 33.43439
## 105 2 5 week2 58 2 52.97460
## 106 2 6 week2 37 2 33.61435
## 107 2 7 week2 41 2 46.15569
## 108 2 8 week2 49 2 58.24270
## 109 2 9 week2 38 2 34.02917
## 110 2 10 week2 31 2 45.42394
## 111 2 11 week2 75 2 75.56273
## 112 2 12 week2 36 2 34.26921
## 113 2 13 week2 60 2 53.13959
## 114 2 14 week2 31 2 38.17088
## 115 2 15 week2 55 2 43.50947
## 116 2 16 week2 35 2 35.31144
## 117 2 17 week2 30 2 43.81039
## 118 2 18 week2 29 2 35.40496
## 119 2 19 week2 26 2 34.55101
## 120 2 20 week2 32 2 33.91569
## 121 1 1 week3 43 3 43.48835
## 122 1 2 week3 55 3 46.38610
## 123 1 3 week3 38 3 39.69818
## 124 1 4 week3 54 3 53.81329
## 125 1 5 week3 65 3 53.74994
## 126 1 6 week3 38 3 37.96412
## 127 1 7 week3 30 3 42.86545
## 128 1 8 week3 38 3 33.33876
## 129 1 9 week3 35 3 33.08816
## 130 1 10 week3 55 3 48.55359
## 131 1 11 week3 41 3 38.14784
## 132 1 12 week3 54 3 46.60136
## 133 1 13 week3 31 3 30.58558
## 134 1 14 week3 34 3 33.09627
## 135 1 15 week3 49 3 47.12745
## 136 1 16 week3 42 3 36.95145
## 137 1 17 week3 38 3 38.49208
## 138 1 18 week3 25 3 30.75406
## 139 1 19 week3 28 3 28.61880
## 140 1 20 week3 25 3 29.53195
## 141 2 1 week3 41 3 50.68318
## 142 2 2 week3 24 3 25.83890
## 143 2 3 week3 28 3 34.85200
## 144 2 4 week3 31 3 31.16397
## 145 2 5 week3 61 3 50.70418
## 146 2 6 week3 33 3 31.34394
## 147 2 7 week3 33 3 43.88527
## 148 2 8 week3 39 3 55.97229
## 149 2 9 week3 27 3 31.75875
## 150 2 10 week3 34 3 43.15353
## 151 2 11 week3 76 3 73.29232
## 152 2 12 week3 27 3 31.99880
## 153 2 13 week3 53 3 50.86917
## 154 2 14 week3 38 3 35.90047
## 155 2 15 week3 55 3 41.23905
## 156 2 16 week3 27 3 33.04102
## 157 2 17 week3 32 3 41.53998
## 158 2 18 week3 33 3 33.13454
## 159 2 19 week3 27 3 32.28059
## 160 2 20 week3 29 3 31.64527
## 161 1 1 week4 41 4 41.21794
## 162 1 2 week4 43 4 44.11568
## 163 1 3 week4 43 4 37.42776
## 164 1 4 week4 56 4 51.54287
## 165 1 5 week4 50 4 51.47953
## 166 1 6 week4 36 4 35.69370
## 167 1 7 week4 27 4 40.59503
## 168 1 8 week4 31 4 31.06834
## 169 1 9 week4 28 4 30.81774
## 170 1 10 week4 53 4 46.28317
## 171 1 11 week4 36 4 35.87743
## 172 1 12 week4 48 4 44.33094
## 173 1 13 week4 25 4 28.31517
## 174 1 14 week4 25 4 30.82585
## 175 1 15 week4 36 4 44.85703
## 176 1 16 week4 31 4 34.68103
## 177 1 17 week4 40 4 36.22166
## 178 1 18 week4 29 4 28.48365
## 179 1 19 week4 29 4 26.34838
## 180 1 20 week4 25 4 27.26154
## 181 2 1 week4 39 4 48.41277
## 182 2 2 week4 20 4 23.56849
## 183 2 3 week4 22 4 32.58159
## 184 2 4 week4 31 4 28.89356
## 185 2 5 week4 49 4 48.43376
## 186 2 6 week4 28 4 29.07352
## 187 2 7 week4 34 4 41.61485
## 188 2 8 week4 55 4 53.70187
## 189 2 9 week4 31 4 29.48833
## 190 2 10 week4 39 4 40.88311
## 191 2 11 week4 66 4 71.02190
## 192 2 12 week4 29 4 29.72838
## 193 2 13 week4 52 4 48.59876
## 194 2 14 week4 35 4 33.63005
## 195 2 15 week4 42 4 38.96864
## 196 2 16 week4 25 4 30.77061
## 197 2 17 week4 46 4 39.26956
## 198 2 18 week4 30 4 30.86412
## 199 2 19 week4 24 4 30.01017
## 200 2 20 week4 25 4 29.37486
## 201 1 1 week5 40 5 38.94752
## 202 1 2 week5 34 5 41.84527
## 203 1 3 week5 28 5 35.15735
## 204 1 4 week5 50 5 49.27246
## 205 1 5 week5 39 5 49.20911
## 206 1 6 week5 29 5 33.42329
## 207 1 7 week5 40 5 38.32462
## 208 1 8 week5 26 5 28.79793
## 209 1 9 week5 22 5 28.54732
## 210 1 10 week5 43 5 44.01275
## 211 1 11 week5 36 5 33.60701
## 212 1 12 week5 43 5 42.06053
## 213 1 13 week5 25 5 26.04475
## 214 1 14 week5 27 5 28.55543
## 215 1 15 week5 32 5 42.58661
## 216 1 16 week5 31 5 32.41061
## 217 1 17 week5 33 5 33.95125
## 218 1 18 week5 28 5 26.21323
## 219 1 19 week5 21 5 24.07797
## 220 1 20 week5 27 5 24.99112
## 221 2 1 week5 38 5 46.14235
## 222 2 2 week5 20 5 21.29807
## 223 2 3 week5 25 5 30.31117
## 224 2 4 week5 26 5 26.62314
## 225 2 5 week5 38 5 46.16335
## 226 2 6 week5 26 5 26.80310
## 227 2 7 week5 37 5 39.34444
## 228 2 8 week5 51 5 51.43145
## 229 2 9 week5 24 5 27.21792
## 230 2 10 week5 34 5 38.61269
## 231 2 11 week5 64 5 68.75148
## 232 2 12 week5 27 5 27.45796
## 233 2 13 week5 32 5 46.32834
## 234 2 14 week5 30 5 31.35963
## 235 2 15 week5 30 5 36.69822
## 236 2 16 week5 22 5 28.50019
## 237 2 17 week5 43 5 36.99914
## 238 2 18 week5 26 5 28.59371
## 239 2 19 week5 32 5 27.73976
## 240 2 20 week5 23 5 27.10444
## 241 1 1 week6 38 6 36.67710
## 242 1 2 week6 28 6 39.57485
## 243 1 3 week6 29 6 32.88693
## 244 1 4 week6 47 6 47.00204
## 245 1 5 week6 32 6 46.93869
## 246 1 6 week6 33 6 31.15287
## 247 1 7 week6 30 6 36.05420
## 248 1 8 week6 26 6 26.52751
## 249 1 9 week6 20 6 26.27691
## 250 1 10 week6 43 6 41.74234
## 251 1 11 week6 38 6 31.33659
## 252 1 12 week6 37 6 39.79011
## 253 1 13 week6 21 6 23.77433
## 254 1 14 week6 25 6 26.28502
## 255 1 15 week6 27 6 40.31620
## 256 1 16 week6 29 6 30.14020
## 257 1 17 week6 27 6 31.68083
## 258 1 18 week6 21 6 23.94281
## 259 1 19 week6 22 6 21.80755
## 260 1 20 week6 21 6 22.72070
## 261 2 1 week6 43 6 43.87193
## 262 2 2 week6 19 6 19.02765
## 263 2 3 week6 24 6 28.04075
## 264 2 4 week6 24 6 24.35272
## 265 2 5 week6 37 6 43.89293
## 266 2 6 week6 27 6 24.53269
## 267 2 7 week6 37 6 37.07402
## 268 2 8 week6 55 6 49.16104
## 269 2 9 week6 22 6 24.94750
## 270 2 10 week6 41 6 36.34228
## 271 2 11 week6 64 6 66.48107
## 272 2 12 week6 21 6 25.18755
## 273 2 13 week6 37 6 44.05792
## 274 2 14 week6 33 6 29.08922
## 275 2 15 week6 26 6 34.42780
## 276 2 16 week6 22 6 26.22977
## 277 2 17 week6 43 6 34.72873
## 278 2 18 week6 36 6 26.32329
## 279 2 19 week6 21 6 25.46934
## 280 2 20 week6 23 6 24.83402
## 281 1 1 week7 47 7 34.40669
## 282 1 2 week7 28 7 37.30443
## 283 1 3 week7 25 7 30.61651
## 284 1 4 week7 42 7 44.73162
## 285 1 5 week7 38 7 44.66828
## 286 1 6 week7 27 7 28.88245
## 287 1 7 week7 31 7 33.78378
## 288 1 8 week7 25 7 24.25709
## 289 1 9 week7 23 7 24.00649
## 290 1 10 week7 39 7 39.47192
## 291 1 11 week7 36 7 29.06618
## 292 1 12 week7 36 7 37.51969
## 293 1 13 week7 19 7 21.50392
## 294 1 14 week7 26 7 24.01460
## 295 1 15 week7 30 7 38.04578
## 296 1 16 week7 26 7 27.86978
## 297 1 17 week7 31 7 29.41041
## 298 1 18 week7 25 7 21.67240
## 299 1 19 week7 23 7 19.53713
## 300 1 20 week7 19 7 20.45029
## 301 2 1 week7 62 7 41.60152
## 302 2 2 week7 18 7 16.75724
## 303 2 3 week7 31 7 25.77034
## 304 2 4 week7 26 7 22.08231
## 305 2 5 week7 36 7 41.62251
## 306 2 6 week7 23 7 22.26227
## 307 2 7 week7 38 7 34.80360
## 308 2 8 week7 59 7 46.89062
## 309 2 9 week7 21 7 22.67708
## 310 2 10 week7 42 7 34.07186
## 311 2 11 week7 60 7 64.21065
## 312 2 12 week7 22 7 22.91713
## 313 2 13 week7 52 7 41.78751
## 314 2 14 week7 30 7 26.81880
## 315 2 15 week7 30 7 32.15739
## 316 2 16 week7 22 7 23.95936
## 317 2 17 week7 43 7 32.45831
## 318 2 18 week7 33 7 24.05287
## 319 2 19 week7 21 7 23.19892
## 320 2 20 week7 23 7 22.56361
## 321 1 1 week8 51 8 32.13627
## 322 1 2 week8 28 8 35.03402
## 323 1 3 week8 24 8 28.34610
## 324 1 4 week8 46 8 42.46121
## 325 1 5 week8 32 8 42.39786
## 326 1 6 week8 25 8 26.61204
## 327 1 7 week8 31 8 31.51337
## 328 1 8 week8 24 8 21.98668
## 329 1 9 week8 21 8 21.73607
## 330 1 10 week8 32 8 37.20150
## 331 1 11 week8 36 8 26.79576
## 332 1 12 week8 31 8 35.24928
## 333 1 13 week8 22 8 19.23350
## 334 1 14 week8 26 8 21.74418
## 335 1 15 week8 37 8 35.77536
## 336 1 16 week8 30 8 25.59936
## 337 1 17 week8 27 8 27.14000
## 338 1 18 week8 20 8 19.40198
## 339 1 19 week8 22 8 17.26672
## 340 1 20 week8 21 8 18.17987
## 341 2 1 week8 50 8 39.33110
## 342 2 2 week8 20 8 14.48682
## 343 2 3 week8 32 8 23.49992
## 344 2 4 week8 23 8 19.81189
## 345 2 5 week8 35 8 39.35210
## 346 2 6 week8 21 8 19.99185
## 347 2 7 week8 35 8 32.53319
## 348 2 8 week8 66 8 44.62020
## 349 2 9 week8 21 8 20.40667
## 350 2 10 week8 39 8 31.80144
## 351 2 11 week8 75 8 61.94023
## 352 2 12 week8 23 8 20.64671
## 353 2 13 week8 28 8 39.51709
## 354 2 14 week8 27 8 24.54838
## 355 2 15 week8 37 8 29.88697
## 356 2 16 week8 22 8 21.68894
## 357 2 17 week8 43 8 30.18789
## 358 2 18 week8 30 8 21.78246
## 359 2 19 week8 21 8 20.92851
## 360 2 20 week8 23 8 20.29319